1. CHUYN LUYN THI I HC 2013 - 2014 KHO ST HM S BIN SON: LU HUY THNG H NI, 8/2013 H V TN: LP :. TRNG :

2. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 1 CHUYN : KHO ST S BIN THIN V V TH HM S CC BI TON LIN QUAN N KHO ST HM S VN 1: TNH N IU CA HM S 1. inh ngha: Hm s f ng bin trn 1 2 1 2 1 2( , , ( ) ( ))K x x K x x f x f x < < Hm s f nghch bin trn 1 2 1 2 1 2( , , ( ) ( ))K x x K x x f x f x < > 2. iu kin cn: Gi s f c o hm trn khong I. a) Nu f ng bin trn khong I th '( ) 0,f x x I b) Nu f nghch bin trn khong I th '( ) 0,f x x I 3.iu kin : Gi s f c o hm trn khong I. a) Nu '( ) 0,f x x I ( '( ) 0f x = ti mt s hu hn im) th f ng bin trn I. b) Nu '( ) 0,f x x I ( '( ) 0f x = ti mt s hu hn im) th f nghch bin trn I. c) Nu '( ) 0,f x x I= , x I th f khng i trn I. Ch : Nu khong I c thay bi on hoc na khong th f phi lin tc trn . Dng ton 1: Xt tnh n iu ca hm s Phng php: xt chiu bin thin ca hm s y = f(x), ta thc hin cc bc nh sau: Tm tp xc nh ca hm s. Tnh y. Tm cc im m ti y = 0 hoc y khng tn ti (gi l cc im ti hn) Lp bng xt du y (bng bin thin). T kt lun cc khong ng bin, nghch bin ca hm s. Bi tp c bn HT 1. Xt tnh n iu ca cc hm s sau: 1) 3 2 2 2y x x x= + 2) 2 (4 )( 1)y x x= 3) 3 2 3 4 1y x x x= + 4) 4 21 2 1 4 y x x= 5) 4 2 2 3y x x= + 6) 4 21 1 2 10 10 y x x= + 7) 2 1 5 x y x = + 8) 1 2 x y x = 9) 1 1 1 y x = 10) 3 2 2y x x= + + 11) 2 1 3y x x= 12) 2 2y x x= www.VNMATH.com 3. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 2 Dng ton2: Tm iu kin hm s lun ng bin hoc nghch bin trn tp xc nh (hoc trn tng khong xc nh) Cho hm s ( , )y f x m= , m l tham s, c tp xc nh D. Hm s f ng bin trn D y 0, x D. Hm s f nghch bin trn D y 0, x D. T suy ra iu kin ca m. Ch : 1) y = 0 ch xy ra ti mt s hu hn im. 2) Nu 2 'y ax bx c= + + th: 0 0 ' 0, 0 0 a b c y x R a = = > 0 0 ' 0, 0 0 a b c y x R a = = < 3) nh l v du ca tam thc bc hai 2 ( )g x ax bx c= + + : Nu < 0 th g(x) lun cng du vi a. Nu = 0 th g(x) lun cng du vi a (tr x = 2 b a ) Nu > 0 th g(x) c hai nghim x1, x2 v trong khong hai nghim th g(x) khc du vi a, ngoi khong hai nghim th g(x) cng du vi a. 4) So snh cc nghim 1 2,x x ca tam thc bc hai 2 ( )g x ax bx c= + + vi s 0: 1 2 0 0 0 0 x x P S >< < > < 1 2 0 0 0 0 x x P S >< < > > 1 20 0x x P< < < 5) hm s 3 2 y ax bx cx d= + + + c di khong ng bin (nghch bin) 1 2( ; )x x bng d th ta thc hin cc bc sau: Tnh y. Tm iu kin hm s c khong ng bin v nghch bin: 0 0 a > (1) Bin i 1 2x x d = thnh 2 2 1 2 1 2( ) 4x x x x d+ = (2) S dng nh l Viet a (2) thnh phng trnh theo m. Gii phng trnh, so vi iu kin (1) chn nghim. Bi tp c bn HT 2. Tm m cc hm s sau lun ng bin trn tp xc nh (hoc tng khong xc nh) ca n: 1) 3 2 3 ( 2)y x mx m x m= + + 2) 3 2 2 1 3 2 x mx y x= + 3) x m y x m + = 4) 4mx y x m + = + HT 3. Tm m hm s: 1) 3 2 3y x x mx m= + + + nghch bin trn mt khong c di bng 1. 2) 3 21 1 2 3 1 3 2 y x mx mx m= + + nghch bin trn mt khong c di bng 3. 4. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 3 3) 3 21 ( 1) ( 3) 4 3 y x m x m x= + + + ng bin trn mt khong c di bng 4. HT 4. Tm m hm s: 1) 3 2 ( 1) ( 1) 1 3 x y m x m x= + + + + ng bin trn khong (1; +). 2) 3 2 3(2 1) (12 5) 2y x m x m x= + + + + ng bin trn khong (2; +). 3) 4 ( 2) mx y m x m + = + ng bin trn khong (1; +). 4) x m y x m + = ng bin trong khong (1; +). BI TP TNG HP NNG CAO HT 5. Cho hm s (1).Tm tt c cc gi tr ca tham s m hm s (1) ng bin trn khong . /s: HT 6. Cho hm s c th (Cm).Tm m hm s ng bin trn khong /s: HT 7. Cho hm s . Tm m hm ng bin trn . /s: 5 4 m HT 8. Cho hm s (1), (m l tham s).Tm m hm s (1) ng bin trn khong (1;2). /s: [ ;1)m HT 9. Cho hm s 3 2 3(2 1) (12 5) 2y x m x m x= + + + + ng bin trn khong ( ; 1) v (2; )+ /s: 7 5 12 12 m HT 10. Cho hm s 3 2 2 (2 7 7) 2( 1)(2 3)y x mx m m x m m= + + . Tm m hm s ng bin trn [2; ).+ /s: 5 1 2 m --------------------------------------------------------- 3 2 3 4y x x mx= + ( ;0) 3m x3 2 2 3(2 1) 6 ( 1) 1y m x m m x= + + + + (2; )+ 1m 3 2 (1 2 ) (2 ) 2y x m x m x m= + + + + ( )0;+ 4 2 2 3 1y x mx m= + www.VNMATH.com 5. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 4 VN 2: CC TR CA HM S I. KIN THC CN NH I.Khi nim cc tr ca hm s Gi s hm s f xc nh trn tp ( )D D v 0x D 1) 0x im cc i ca f nu tn ti khong ( ; )a b D v 0 ( ; )x a b sao cho 0( ) ( )f x f x< , { }0( ; )x a b x . Khi 0( )f x c gi l gi tr cc i (cc i) ca f . 2) 0x im cc tiu ca f nu tn ti khong ( ; )a b D v 0 ( ; )x a b sao cho 0( ) ( )f x f x> , { }0( ; )x a b x . Khi 0( )f x c gi l gi tr cc tiu (cc tiu) ca f . 3) Nu 0x l im cc tr ca f th im 0 0( ; ( ))x f x c gi l im cc tr ca th hm s f . II. iu kin cn hm s c cc tr Nu hm s f c o hm ti 0x v t cc tr ti im th 0'( ) 0f x = . Ch : Hm s f ch c th t cc tr ti nhng im m ti o hm bng 0 hoc khng c o hm. III. iu kin hm s c cc tr 1. nh l 1: Gi s hm s f lin tc trn khong ( ; )a b cha im 0x v c o hm trn { }( ; ) oa b x 1) Nu '( )f x i du t m sang dng khi x i qua 0x th f t cc tiu ti 0x . 2) Nu '( )f x i du t dng sang m khi x i qua 0x th f t cc i ti 0x 2. nh l 2: Gi s hm s f c o hm trn khong ( ; )a b cha im 0x , 0'( ) 0f x = v c o hm cp hai khc 0 ti im 0x . 1) Nu 0"( ) 0f x < th f t cc i ti 0x . 2) Nu 0"( ) 0f x > th f t cc tiu ti 0x . II. CC DNG TON Dng ton 1: Tm cc tr ca hm s Qui tc 1: Dng nh l 1. Tm '( )f x . Tm cc im ( 1,2,...)ix i = m ti o hm bng 0 hoc khng c o hm. Xt du '( )f x . Nu '( )f x i du khix i qua ix th hm s t cc tr ti ix . Qui tc 2: Dng nh l 2. Tnh '( )f x Gii phng trnh '( ) 0f x = tm cc nghim ( 1,2,...)ix i = Tnh "( )f x v "( ) ( 1,2,...)if x i = . Nu "( ) 0if x < th hm s t cc i ti ix . Nu "( ) 0if x > th hm s t cc tiu ti ix 6. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 5 Bi tp c bn HT 11. Tm cc tr ca cc hm s sau: 1) 2 3 3 2y x x= 2) 3 2 2 2 1y x x x= + 3) 3 21 4 15 3 y x x x= + 4) 4 2 3 2 x y x= + 5) 4 2 4 5y x x= + 6) 4 2 3 2 2 x y x= + + 7) 2 3 6 2 x x y x + + = + 8) 2 3 4 5 1 x x y x + + = + 9) 2 2 15 3 x x y x = 10) 3 4 ( 2) ( 1)y x x= + 11) 2 2 4 2 1 2 3 x x y x x + = + 12) 2 2 3 4 4 1 x x y x x + + = + + 13) 2 4y x x= 14) 2 2 5y x x= + 15) 2 2y x x x= + Dng ton 2: Tm iu kin hm s c cc tr 1. Nu hm s ( )y f x= t cc tr ti im 0x th 0'( ) 0f x = hoc ti 0x khng c o hm. 2. hm s ( )y f x= ) t cc tr ti im 0x th '( )f x i du khi x i qua 0x . Ch : Hm s bc ba 3 2 y ax bx cx d= + + + c cc tr Phng trnh ' 0y = c hai nghim phn bit. Khi nu x0 l im cc tr th ta c th tnh gi tr cc tr y(x0) bng hai cch: + 3 2 0 0 0 0( )y x ax bx cx d= + + + + 0 0( )y x Ax B= + , trong Ax + B l phn d trong php chia y cho y. Bi tp c bn HT 12. Tm m hm s: 1) 3 2 ( 2) 3 5y m x x mx= + + + c cc i, cc tiu. 2) 3 2 2 3( 1) (2 3 2) ( 1)y x m x m m x m m= + + c cc i, cc tiu. 3) 3 2 2 3 3 3( 1)y x mx m x m= + 4) 3 2 2 3(2 1) 6 ( 1) 1y x m x m m x= + + + + 2x = 5) 3 2 2 3 ( 1) 2y x mx m x= + + t cc i ti 6) 4 2 2( 2) 5y mx m x m= + + c mt cc i 1 . 2 x = www.VNMATH.com 7. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 6 HT 13. Tm , , ,a b c d hm s: 1) 3 2 y ax bx cx d= + + + t cc tiu bng 0 ti 0x = v t cc i bng 4 27 ti 1 3 x = 2) 4 2 y ax bx c= + + c th i qua gc to O v t cc tr bng 9 ti 3x = . HT 14. Tm m cc hm s sau khng c cc tr: 1) 3 2 3 3 3 4y x x mx m= + + + 2) 3 2 3 ( 1) 1y mx mx m x= + HT 15. Tm m hm s : 1) 3 2 2 2 2( 1) ( 4 1) 2( 1)y x m x m m x m= + + + + t cc tr ti hai im 1 2,x x sao cho: 1 2 1 2 1 1 1 ( ) 2 x x x x + = + . 2) 3 21 1 3 y x mx mx= + t cc tr ti hai im 1 2,x x 2 sao cho: 1 2 8x x . 3) 3 21 1 ( 1) 3( 2) 3 3 y mx m x m x= + + t cc tr ti hai im 1 2,x x sao cho: 1 22 1x x+ = . HT 16. Tm m th hm s : 1) 3 2 4y x mx= + c hai im cc tr l A, B v 2 2 900 729 m AB = . 2) 4 2 4y x mx x m= + + c 3 im cc tr l A, B, C v tam gic ABC nhn gc to O lm trng tm. BI TP TNG HP V NNG CAO HT 17. Tm m th hm s : 1) 3 2 2 12 13y x mx x= + c hai im cc tr cch u trc tung. /s: 0m = 2) 3 2 3 3 4y x mx m= + c cc im cc i, cc tiu i xng nhau qua ng phn gic th nht. /s: 1 2 m = 3) 3 2 3 3 4y x mx m= + c cc im cc i, cc tiu v mt pha i vi ng thng : 3 2 8 0d x y + = . /s: { 4 ;10} 3 m HT 18. Tm m th hm s: 1) 3 2 3y x x m= + + c 2 im cc tr ti A, B sao cho 0 120AOB = /s: 12 132 0, 3 m m + = = 2) 4 2 2 2y x mx= + c 3 im cc tr to thnh 1 tam gic c ng trn ngoi tip i qua 3 9 ; 5 5 D /s: 1m = 8. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 7 3) 4 2 2 2y x mx m m= + + + c 3 im cc tr to thnh 1 tam gic c mt gc bng 0 120 . /s: 3 1 3 m = 4) 4 2 4 2 2y x mx m m= + + c 3 im cc tr to thnh 1 tam gic c din tch bng 4. /s: 3 2m = HT 19. Tm m hm s: 1) 3 3 2y x mx= + c hai im cc tr v ng trn qua 2 im cc tr ct ng trn tm (1;1)I bn knh bng 1 ti hai im A, B sao cho din tch tam gic IAB ln nht. /s: 2 3 2 m = 2) 3 2 4 3y x mx x= + c hai im cc tr 1 2,x x tha mn: 1 24 0x x+ = /s: 9 2 m = HT 20. Tm m hm s: 1) 3 2 2 3( 1) 6( 2) 1y x m x m x= + + c ng thng i qua hai im cc tr song song vi ng thng 4 1y x= . /s: 5m = 2) 3 2 2 3( 1) 6 (1 2 )y x m x m m x= + + c cc im cc i, cc tiu ca th nm trn ng thng 4y x= . /s: 1m = 3) 3 2 7 3y x mx x= + + + c ng thng i qua cc im cc i, cc tiu vung gc vi ng thng 3 7y x= . /s: 3 10 2 m = 4) 3 2 2 3y x x m x m= + + c cc im cc i v cc tiu i xng nhau qua ng thng (): 1 5 2 2 y x= . /s: 0m = ------------------------------------------------------- www.VNMATH.com 9. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 8 VN 3: KHO ST S BIN THIN V V TH HM S I. KIN THC CN NH 1. Cc bc kho st s bin thin v v th ca hm s Tm tp xc nh ca hm s. Xt s bin thin ca hm s: + Tm cc gii hn ti v cc, gii hn v cc v tm tim cn (nu c). + Tnh 'y . + Tm cc im ti o hm ' 0y = hoc khng xc nh. + Lp bng bin thin ghi r du ca o hm, chiu bin thin, cc tr ca hm s. V th ca hm s: + Tm im un ca th (i vi hm s bc ba v hm s trng phng). + V cc ng tim cn (nu c) ca th. + Xc nh mt s im c bit ca th nh giao im ca th vi cc trc to (trong trng hp th khng ct cc trc to hoc vic tm to giao im phc tp th c th b qua). C th tm thm mt s im thuc th c th v chnh xc hn. 2. Kho st s bin thin v v th hm bc ba 3 2 ( 0)y ax bx cx d a= + + + Tp xc nh D = . th lun c mt im un v nhn im un lm tm i xng. Cc dng th: a > 0 a < 0 ' 0y = c 2 nghim phn bit 2 ' 3 0b ac = > ' 0y = c nghim kp 2 ' 3 0b ac = = ' 0y = v nghim 2 ' 3 0b ac = < y x0 I y x0 I y x0 I y x0 I 10. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 9 3. Hm s trng phng 4 2 ( 0)y ax bx c a= + + Tp xc nh D = th lun nhn trc tung lm trc i xng. Cc dng th: 4. Hm s nht bin ( 0; 0) ax b y c ad bc cx d + = + Tp xc nh D =d c th c mt tim cn ng l v mt tim cn ngang l . Giao im ca hai tim cn l tm i xng ca th hm s. Cc dng th: Bi tp c bn HT 21. Kho st s bin thin v v th cc hm s sau: 1. 3 2 3 1y x x= + 2. 3 2 1 3 x y x x= + 3. 3 2 2 1 3 x y x x= + + 4. 4 2 2 2y x x= + 5. 4 2 1y x x= + 6. 1 1 x y x = + 7. 2 1 1 x y x = 8. 1 2 1 x y x = + ---------------------------------------------------- d x c = a y c = a > 0 a < 0 c 3 nghim phn bit ch c 1 nghim y x0 y x0 y x0 y x0 0 ad bc > x y 0 ad bc < x y www.VNMATH.com 11. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 10 VN 4: S TNG GIAO CA CC TH Dng ton 1: Dng th hm s bin lun s nghim phng trnh C s ca phng php: Xt phng trnh: ( ) ( )f x g x= (1) S nghim ca phng trnh (1) = S giao im ca 1( ) : ( )C y f x= v 2( ) : ( )C y g x= Nghim ca phng trnh (1) l honh giao im ca 1( ) : ( )C y f x= v 2( ) : ( )C y g x= bin lun s nghim ca phng trnh ( , ) 0F x m = (*) bng th ta bin i (*) v dng sau: ( , ) 0 ( ) ( )F x m f x g m= = (1) Khi (1) c th xem l phng trnh honh giao im ca hai ng: ( ) : ( )C y f x= v : ( )d y g m= d l ng thng cng phng vi trc honh. Da vo th (C) ta bin lun s giao im ca (C) v d . T suy ra s nghim ca (1) Bi tp c bn HT 22. Kho st s bin thin v v th (C) ca hm s. Dng th (C) bin lun theo m s nghim ca phng trnh: 1) 3 3 3 1; 3 1 0y x x x x m= + + = 2) 3 3 3 1; 3 1 0y x x x x m= + + + = 3) 3 3 2 3 1; 3 2 2 0y x x x x m m= + = 4) 3 3 3 1; 3 4 0y x x x x m= + + + = 5) 4 2 4 2 2 2; 4 4 2 0 2 x y x x x m= + + + = 6) 4 2 4 2 2 2; 2 2 0y x x x x m= + + = HT 23. Kho st s bin thin v v th (C) ca hm s. Dng th (C) bin lun theo m s nghim ca phng trnh: 1) 3 2 3 2 ( ) : 3 6; 3 6 3 0C y x x x x m= + + + = 2) 33 2 2 ( ) : 2 9 12 4; 2 9 12 0C y x x x x x x m= + + + = 3) 2 2 2 ( ) : ( 1) (2 ); ( 1) 2 ( 1) (2 )C y x x x x m m= + + = + 4) 1 11 1 1 ( ) : ; ; ; 1 1 1 1 1 x xx x x C y m m m m x x x x x = = = = = + + + + + ------------------------------------------------ y x g(m A (C) (4) : y = g(m)yC yCT xA 12. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 11 Dng ton 2: Tm iu kin tng giao gia cc th 1.Cho hai th 1( ) : ( )C y f x= v 2( ) : ( )C y g x= . tm honh giao im ca 1( )C v 2( )C ta gii phng trnh: ( ) ( )f x g x= (*) (gi l phng trnh honh giao im). S nghim ca phng trnh (*) bng s giao im ca hai th. 2. th hm s bc ba 3 2 ( 0)y ax bx cx d a= + + + ct trc honh ti 3 im phn bit Phng trnh 3 2 0ax bx cx d+ + + = c 3 nghim phn bit. Bi tp c bn HT 24. Tm to giao im ca cc th ca cc hm s sau: 1) 2 3 3 2 2 1 2 2 x y x x y = + = + 2) 2 2 4 1 2 4 x y x y x x = = + + 3) 3 4 3 2 y x x y x = = + HT 25. Tm m th cc hm s: 1) 2 2 ( 1)( 3)y x x mx m= + ct trc honh ti ba im phn bit. 2) 3 2 3 (1 2 ) 1y mx mx m x= + ct trc honh ti ba im phn bit. 3) 3 2 2 2 ; 2y x x mx m y x= + + + = + ct nhau ti ba im phn bit. 4) 3 2 2 2 2 2 1; 2 2y x x x m y x x= + + = + ct nhau ti ba im phn bit. HT 26. Tm m th cc hm s: 1) 4 2 2 1;y x x y m= = ct nhau ti bn im phn bit. 2) 4 2 3 ( 1)y x m m x m= + + ct trc honh ti bn im phn bit. 3) 4 2 2 (2 3) 3y x m x m m= + ct trc honh ti bn im phn bit. HT 27. Bin lun theo m s giao im ca cc th ca cc hm s sau: 1) 3 3 2 ( 2) y x x y m x = = 2) 3 3 3 ( 3) x y x y m x = + = HT 28. Tm m th ca cc hm s: 1) 3 1 ; 2 4 x y y x m x + = = + ct nhau ti hai im phn bit A, B. Khi tm m on AB ngn nht. 2) 4 1 ; 2 x y y x m x = = + ct nhau ti hai im phn bit A, B. Khi tm m on AB ngn nht. www.VNMATH.com 13. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 12 BI TP TNG HP V NNG CAO HT 29. Tm m hm s: 1) 2 1 ( ) 1 x y C x = + ct ng thng : y x m = + ti hai im phn bit A, B sao cho 2 2AB = /s: 1; 7m m= = 2) 1 ( ) 2 x y C x = ct ng thng : y x m = + ti hai im phn bit A, B sao cho A, B c di nh nht. /s: 1 2 m = 3) 2 1 ( ) 1 x y C x = ct ng thng : y x m = + ti hai im phn bit A, B sao cho tam gic OAB vung ti O. /s: 2m = 4) 2 2 3 ( ) 2 mx m y C x = + ct ng thng : 2y x = ti hai im phn bit A, B sao cho 0 45AOB = 5) (1 ) 2(1 )m x m y x + + = ct ng thng : y x = ti hai im phn bit A, B sao cho: 4 OA OB OB OA + = 6) 3 1 1 x y x + = ct ng thng : ( 1) 2y m x m = + + ti hai im phn bit A, B sao cho tam gic OAB c din tch bng 3 . 2 7) 1 ( ) 2 1 x y C x + = + ct ng thng : 2 2 1 0,mx y m + + = ct th (C) ti hai im phn bit A, B sao cho biu thc 2 2 P OA OB= + t gi tr nh nht. HT 30. Cho hm s 2 ( ) 1 x y C x + = Gi I l giao im ca hai tim cn. Tm trn th (C) hai im A, B sao cho tam gic IAB nhn (4; 2)H lm trc tm. /s: (2;4),( 2;0) HT 31. Cho hm s 1 ( ) 2 1 x y C x + = Xc nh m ng thng : y x m = + ct th (C) ti hai im phn bit c honh 1 2,x x sao cho tng 1 2'( ) '( )f x f x+ t gi tr ln nht. HT 32. Cho hm s 1 ( ) 2 1 x y C x = + Xc nh m ng thng : y x m = + ct th (C) ti hai im phn bit c honh 1 2,x x sao cho tng 1 2'( ) '( )f x f x+ t gi tr nh nht. HT 33. Cho hm s 3 4 ( ) 2 3 x y C x = Xc nh ta cc im trn th (C) sao cho tng khong cch t im n trc honh gp 2 ln khong cch t im n tim cn ng. ----------------------------------------------------- 14. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 13 VN 5: S TIP XC CA HAI NG CONG 1. ngha hnh hc ca o hm: o hm ca hm s ( )y f x= ti im 0x l h s gc ca tip tuyn vi th (C) ca hm s ti im ( )0 0 0; ( )M x f x . Khi phng trnh tip tuyn ca (C) ti im ( )0 0 0; ( )M x f x l: 0 0 0'( )( )y y f x x x = 0 0( ( ))y f x= 2. iu kin cn v hai ng 1( ) : ( )C y f x= v 2( ) : ( )C y g x= tip xc nhau l h phng trnh sau c nghim: ( ) ( ) '( ) '( ) f x g x f x g x = = (*) Nghim ca h (*) l honh ca tip im ca hai ng . 3. Nu 1( ) :C y px q= + v 2 2( ) :C y ax bx c= + + th (C1) v (C2) tip xc nhau phng trnh 2 ax bx c px q+ + = + c nghim kp. Dng ton 1: Lp phng trnh tip tuyn ca ng cong (C): y = f(x) Bi ton 1: Vit phng trnh tip tuyn ca( ) : ( )C y f x= ti im ( )0 0 0;M x y : Nu cho 0x th tm 0 0( )y f x= Nu cho 0y th tm 0x l nghim ca phng trnh 0( )f x y= . Tnh ' '( )y f x= . Suy ra 0 0'( ) '( )y x f x= . Phng trnh tip tuyn l: 0 0 0'( )( )y y f x x x = Bi ton 2: Vit phng trnh tip tuyn ca ( ) : ( )C y f x= bit c h s gc k cho trc. Cch 1: Tm to tip im. Gi M(x0; y0) l tip im. Tnh f (x0). c h s gc k f (x0) = k (1) Gii phng trnh (1), tm c x0 v tnh y0 = f(x0). T vit phng trnh ca . Cch 2: Dng iu kin tip xc. Phng trnh ng thng c dng: y = kx + m. tip xc vi (C) khi v ch khi h phng trnh sau c nghim: ( ) '( ) f x kx m f x k = + = (*) www.VNMATH.com 15. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 14 Gii h (*), tm c m. T vit phng trnh ca . Ch : H s gc k ca tip tuyn c th c cho gin tip nh sau: + to vi chiu dng trc honh gc th k = tan + song song vi ng thng d: y = ax + b th k = a + vung gc vi ng thng d: y = ax + b (a 0) th k = 1 a + to vi ng thng d: y = ax + b mt gc th tan 1 k a ka = + Bi ton 3: Vit phng trnh tip tuyn ca (C): y = f(x), bit i qua im ( ; )A AA x y . Cch 1:Tm to tip im. Gi M(x0; y0) l tip im. Khi : y0 = f(x0), y0 = f (x0). Phng trnh tip tuyn ti M: y y0 = f (x0).(x x0) i qua ( ; )A AA x y nn: yA y0 = f (x0).(xA x0) (2) Gii phng trnh (2), tm c x0. T vit phng trnh ca . Cch 2: Dng iu kin tip xc. Phng trnh ng thng i qua ( ; )A AA x y v c h s gc k: y yA = k(x x1) tip xc vi (C) khi v ch khi h phng trnh sau c nghim: ( ) ( ) '( ) A Af x k x x y f x k = + = (*) Gii h (*), tm c x (suy ra k). T vit phng trnh tip tuyn . Bi tp c bn HT 34. Vit phng trnh tip tuyn ca (C) ti im c ch ra: 1) 3 2 ( ) : 3 7 1C y x x x= + ti A(0; 1) 2) ( ) :C 4 2 2 1y x x= + ti B(1; 0) 3) (C): 3 4 2 3 x y x + = ti C(1; 7) 4) (C): 1 2 x y x + = ti cc giao im ca (C) vi trc honh, trc tung. 5) (C): 2 2 2 1y x x= + ti cc giao im ca (C) vi trc honh, trc tung. 6) (C): 3 3 1y x x= + ti im un ca (C). HT 35. Vit phng trnh tip tuyn ca (C) ti cc giao im ca (C) vi ng c ch ra: 1) (C): 3 2 2 3 9 4y x x x= + v d: 7 4y x= + . 16. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 15 2) (C): 3 2 2 3 9 4y x x x= + v (P): 2 8 3y x x= + . HT 36. Tnh din tch tam gic chn hai trc to bi tip tuyn ca th (C) ti im c ch ra: (C): 5 11 2 3 x y x + = ti im A c xA = 2 . HT 37. Tm m tip tuyn ca th (C) ti im c ch ra chn hai trc to mt tam gic c din tch bng S cho trc: 1) (C): 2 1 x m y x + = ti im A c xA = 2 v 1 2 S = . 2) (C): 3 2 x m y x = + ti im B c xB = 1 v S = 9 2 . 3) (C): 3 1 ( 1)y x m x= + + ti im C c xC = 0 v S = 8. HT 38. Vit phng trnh tip tuyn ca (C), bit c h s gc k c ch ra: 1) (C): 3 2 2 3 5y x x= + ; k = 12 2) (C): 2 1 2 x y x = ; k = 3 HT 39. Vit phng trnh tip tuyn ca (C), bit song song vi ng thng d cho trc: 1) (C): 3 2 2 3 1 3 x y x x= + + ; d: y = 3x + 2 2) (C): 2 1 2 x y x = ; d: 3 2 4 y x= + HT 40. Vit phng trnh tip tuyn ca (C), bit vung gc vi ng thng d cho trc: 1) (C): 3 2 2 3 1 3 x y x x= + + ; d: 2 8 x y = + 2) (C): 2 1 2 x y x = ; d: y x= HT 41. Tm m tip tuyn ca (C) ti im c ch ra song song vi ng thng d cho trc: 1) (C): 2 (3 1) ( 0) m x m m y m x m + + = + ti im A c yA = 0 v d: 10y x= . HT 42. Vit phng trnh tip tuyn ca (C), bit i qua im c ch ra: 1) (C): 3 3 2y x x= + ; A(2; 4) 2) (C): 3 3 1y x x= + ; B(1; 6) 3) (C): ( ) 2 2 2y x= ; C(0; 4) 4) (C): 4 21 3 3 2 2 y x x= + ; 3 0; 2 D 5) (C): 2 2 x y x + = ; E(6; 5) 6) (C): 3 4 1 x y x + = ; F(2; 3) HT 43. Tm m hai ng (C1), (C2) tip xc nhau: 1) 3 2 1 2( ) : (3 ) 2; ( ) :C y x m x mx C= + + + trc honh 2) 3 2 1 2( ) : 2 ( 1) ; ( ) :C y x x m x m C= + trc honh 3) 3 1 2( ) : ( 1) 1; ( ) : 1C y x m x C y x= + + + = + 4) 3 2 1 2( ) : 2 2 1; ( ) :C y x x x C y x m= + + = + HT 44. Tm m hai ng (C1), (C2) tip xc nhau: 1) 4 2 2 1 2( ) : 2 1; ( ) : 2C y x x C y mx m= + + = + www.VNMATH.com 17. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 16 2) 4 2 2 1 2( ) : 1; ( ) :C y x x C y x m= + = + 3) 4 2 2 1 2 1 9 ( ) : 2 ; ( ) : 4 4 C y x x C y x m= + + = + 4) 2 2 2 1 2( ) : ( 1) ( 1) ; ( ) : 2C y x x C y x m= + = + 5) 2 1 2 (2 1) ( ) : ; ( ) : 1 m x m C y C y x x = = Dng ton 2: Tm nhng im trn ng thng d m t c th v c 1, 2, 3, tip tuyn vi th (C): ( )y f x= Gi s d: ax + by +c = 0. M(xM; yM) d. Phng trnh ng thng qua M c h s gc k: y = k(x xM) + yM tip xc vi (C) khi h sau c nghim: ( ) ( ) (1) '( ) (2) M Mf x k x x y f x k = + = Th k t (2) vo (1) ta c: f(x) = (x xM).f (x) + yM (C) S tip tuyn ca (C) v t M = S nghim x ca (C) Bi tp c bn HT 45. Tm cc im trn th (C) m t v c ng mt tip tuyn vi (C): 1) 3 2 ( ) : 3 2C y x x= + 2) 3 ( ) : 3 1C y x x= + HT 46. Tm cc im trn ng thng d m t v c ng mt tip tuyn vi (C): 1) 1 ( ) : 1 x C y x + = ; d l trc tung 2) 3 ( ) : 1 x C y x + = ; d: y = 2x + 1 HT 47. Tm cc im trn ng thng d m t v c t nht mt tip tuyn vi (C): 1) 2 1 ( ) : 2 x C y x + = ; d: x = 3 2) 3 4 ( ) : 4 3 x C y x + = ; d: y = 2 HT 48. Tm cc im trn ng thng d m t v c ba tip tuyn vi (C): 1) 3 2 ( ) : 3 2C y x x= + ; d: y = 2 2) 3 ( ) : 3C y x x= ; d: x = 2 3) 3 ( ) : 3 2C y x x= + + ; d l trc honh 4) 3 ( ) : 12 12C y x x= + ; d: y = 4 HT 49. T im A c th k c bao nhiu tip tuyn vi (C): 1) 3 2 ( ) : 9 17 2C y x x x= + + ; A(2; 5) 2) 3 21 4 4 ( ) : 2 3 4; ; 3 9 3 C y x x x A = + + HT 50. T mt im bt k trn ng thng d c th k c bao nhiu tip tuyn vi (C): 1) 3 2 ( ) : 6 9 1C y x x x= + ; : 2d x = 2) 3 ( ) : 3C y x x= ; : 2d x = 18. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 17 Dng ton 3: Tm nhng im m t c th v c 2 tip tuyn vi th (C): y = f(x) v 2 tip tuyn vung gc vi nhau Gi M(xM; yM). Phng trnh ng thng qua M c h s gc k: y = k(x xM) + yM tip xc vi (C) khi h sau c nghim: ( ) ( ) (1) '( ) (2) M Mf x k x x y f x k = + = Th k t (2) vo (1) ta c: f(x) = (x xM).f (x) + yM (C) Qua M v c 2 tip tuyn vi (C) (C) c 2 nghim phn bit x1, x2. Hai tip tuyn vung gc vi nhau f (x1).f (x2) = 1 T tm c M. Ch : Qua M v c 2 tip tuyn vi (C) sao cho 2 tip im nm v hai pha vi trc honh th 1 2 (3) 2 ( ). ( ) 0 co nghiem phan biet f x f x < Bi tp c bn HT 51. Chng minh rng t im A lun k c hai tip tuyn vi (C) vung gc vi nhau. Vit phng trnh cc tip tuyn : 2 1 ( ) : 2 3 1; 0; 4 C y x x A = + HT 52. Tm cc im trn ng thng d m t c th v c hai tip tuyn vi (C) vung gc vi nhau: 1) 3 2 ( ) : 3 2C y x x= + ; d: y = 2 2) 3 2 ( ) : 3C y x x= + ; d l trc honh Dng ton 4: Cc bi ton khc v tip tuyn HT 53. Cho hypebol (H) v im M bt k thuc (H). Gi I l giao im ca hai tim cn. Tip tuyn ti M ct 2 tim cn ti A v B. 1) Chng minh M l trung im ca on AB. 2) Chng minh din tch ca IAB l mt hng s. 3) Tm im M chu vi IAB l nh nht. 4) Tm M bn knh, chu vi, din tch ng trn ngoi tip tam gic IAB t gi tr nh nht. 5) Tm M bn knh, chu vi, din tch ng trn ni tip tam gic IAB t gi tr ln nht. 6) Tm M khong cch t I n tip tuyn l ln nht. 1) 2 1 ( ) : 1 x H y x = 2) 1 ( ) : 1 x H y x + = 3) 4 5 ( ) : 2 3 x H y x = + HT 54. Tm m tip tuyn ti im M bt k thuc hypebol (H) ct hai ng tim cn to thnh mt tam gic c din tch bng S: 1) 2 3 ( ) : ; 8 mx H y S x m + = = www.VNMATH.com 19. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 18 VN 7: KHONG CCH Kin thc c bn: 1) Khong cch gia hai im A, B: AB = 2 2 ( ) ( )B A B Ax x y y + 2) Khong cch t im M(x0; y0) n ng thng : ax + by + c = 0: d(M, ) = 0 0 2 2 ax by c a b + + + 3) Din tch tam gic ABC: S = ( ) 2 2 21 1 . .sin . . 2 2 AB AC A AB AC AB AC= Bi tp c bn HT 55. Tm cc im M thuc hypebol (H) sao cho tng cc khong cch t n hai tim cn l nh nht. 1) 2 ( ) : 2 x H y x + = 2) 2 1 ( ) : 1 x H y x = + 3) 4 9 ( ) : 3 x H y x = HT 56. Tm cc im M thuc hypebol (H) sao cho tng cc khong cch t n hai trc to l nh nht. 1) 1 ( ) : 1 x H y x = + 2) 2 1 ( ) : 2 x H y x + = 3) 4 9 ( ) : 3 x H y x = HT 57. Cho hypebol (H). Tm hai im A, B thuc hai nhnh khc nhau ca (H) sao cho di AB l nh nht. 1) 1 ( ) : 1 x H y x = + 2) 2 3 ( ) : 2 x H y x + = 3) 4 9 ( ) : 3 x H y x = HT 58. Cho (C) v ng thng d. Tm m d ct (C) ti 2 im A, B sao cho di AB l nh nht. 1 ( ) : ; : 2 0 1 x H y d x y m x + = + = 20. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 19 N TP TNG HP PHN I: TNH N IU CA HM S HT 1. Cho hm s 3 21 ( 1) (3 2) 3 y m x mx m x= + + (1).Tm tt c cc gi tr ca tham s m hm s (1) ng bin trn tp xc nh ca n. /s: 2m HT 2. Cho hm s 3 2 3 4y x x mx= + (1).Tm tt c cc gi tr ca tham s m hm s (1) ng bin trn khong ( ; 0) . /s: 3m HT 3. Cho hm s x3 2 2 3(2 1) 6 ( 1) 1y m x m m x= + + + + c th (Cm).Tm m hm s ng bin trn khong (2; )+ /s: 1m HT 4. Cho hm s 3 2 (1 2 ) (2 ) 2y x m x m x m= + + + + . Tm m hm ng bin trn ( )0;+ . /s: 5 4 m HT 5. Cho hm s 4 2 2 3 1y x mx m= + (1), (m l tham s).Tm m hm s (1) ng bin trn khong (1; 2). /s: ( ;1m . HT 6. Cho hm s 4mx y x m + = + (1). Tm tt c cc gi tr ca tham s m hm s (1) nghch bin trn khong ( ;1) ./s: 2 1m < . HT 7. Cho hm s 3 2 3y x x mx m= + + + . Tm m hm s nghch bin trn on c di bng 1. /s: 9 4 m = PHN II: CC TR CA HM S HT 8. Cho hm s 3 2 (1 2 ) (2 ) 2y x m x m x m= + + + + (m l tham s) (1). Tm cc gi tr ca m th hm s (1) c im cc i, im cc tiu, ng thi honh ca im cc tiu nh hn 1. /s: 5 7 4 5 m< < . HT 9. Cho hm s 3 2 ( 2) 3 5y m x x mx= + + + , m l tham s.Tm cc gi tr ca m cc im cc i, cc tiu ca th hm s cho c honh l cc s dng. /s: 3 2m < < HT 10. Cho hm s 3 2 3 2 3( 2) 6(5 1) (4 2).y x m x m x m= + + + + Tm m hm s t cc tiu ti (0 1;2x /s: 1 0 3 m < HT 11. Cho hm s 4 21 3 2 2 y x mx= + (1).Xc nh m th ca hm s (1) c cc tiu m khng c cc i. /s: 0m HT 12. Cho hm s 4 2 2 4 ( ).my x mx C= + Tm cc gi tr ca m tt c cc im cc tr ca ( )mC u nm trn cc trc ta . /s: 2; 0m m= HT 13. Cho hm s 3 2 2 (2 1) ( 3 2) 4y x m x m m x= + + + (m l tham s) c th l (Cm). Xc nh m (Cm) c cc im cc i v cc tiu nm v hai pha ca trc tung. /s:1 2m< < . HT 14. Cho hm s 3 21 (2 1) 3 3 y x mx m x= + (m l tham s) c th l (Cm). Xc nh m (Cm) c cc im cc i, cc tiu nm v cng mt pha i vi trc tung. /s: 1 1 2 m m > HT 15. Cho hm s 3 2 3 2y x x mx m= + + + (m l tham s) c th l (Cm). Xc nh m (Cm) c cc im cc i www.VNMATH.com 21. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 20 v cc tiu nm v hai pha i vi trc honh. /s: 3m < HT 16. Cho hm s 3 2 31 4 ( 1) ( 1) ( ). 3 3 y x m x m C= + + + Tm m cc im cc tr ca hm s (C) nm v hai pha (pha trong v pha ngoi) ca ng trn c phng trnh: 2 2 4 3 0.x y x+ + = /s: 1 2 m < HT 17. Cho hm s 3 2 3 3 4y x mx m= + (m l tham s) c th l (Cm). Xc nh m (Cm) c cc im cc i v cc tiu i xng nhau qua ng thng y = x. /s: 2 2 m = HT 18. Cho hm s 3 2 3 3 1y x mx m= + . Vi gi tr no ca m th th hm s c im cc i v im cc tiu i xng vi nhau qua ng thng : 8 74 0d x y+ = . /s: 2m = HT 19. Cho hm s 3 2 2 3 2 3 3(1 )y x mx m x m m= + + + (1).Vit phng trnh ng thng qua hai im cc tr ca th hm s (1). /s: 2 2y x m m= + . HT 20. Cho hm s 3 2 3 2 ( ).my x x mx C= + + Tm m ( )mC c cc i v cc tiu, ng thi cc im cc tr ca hm s cch u ng thng : 1 0.d x y = /s: 0m = HT 21. Cho hm s 3 2 3 2y x x mx= + (m l tham s) c th l (Cm). Xc nh m (Cm) c cc im cc i v cc tiu cch u ng thng 1y x= . /s: 3 0; 2 m = HT 22. Cho hm s 3 2 3y x x mx= + (1). Vi gi tr no ca m th th hm s (1) c cc im cc i v im cc tiu i xng vi nhau qua ng thng : 2 5 0d x y = . /s: 0m = HT 23. Cho hm s 3 2 3( 1) 9 2y x m x x m= + + + (1) c th l (Cm). Vi gi tr no ca m th th hm s c im cc i v im cc tiu i xng vi nhau qua ng thng 1 : 2 d y x= . /s: 1m = . HT 24. Cho hm s 3 21 1 ( 1) 3( 2) 3 3 y x m x m x= + + , vi m l tham s thc. Xc nh m hm s cho t cc tr ti 1 2,x x sao cho 1 22 1x x+ = . /s: 4 34 4 m = . HT 25. Cho hm s 3 2 3( 1) 9y x m x x m= + + , vi m l tham s thc. Xc nh m hm s cho t cc tr ti 1 2,x x sao cho 1 2 2x x ./s: 3 1 3m < v 1 3 1.m + < HT 26. Cho hm s 3 2 (1 2 ) (2 ) 2y x m x m x m= + + + + , vi m l tham s thc. Xc nh m hm s cho t cc tr ti 1 2,x x sao cho 1 2 1 3 x x > ./s: 3 29 1 8 m m + > < HT 27. Cho hm s 3 2 4 3y x mx x= + . Tm m hm s c hai im cc tr 1 2,x x tha 1 24x x= . /s: 9 2 m = HT 28. Tm cc gi tr ca m hm s 3 2 21 1 ( 3) 3 2 y x mx m x= + c cc i 1x , cc tiu 2x ng thi 1x ; 2x l di cc cnh gc vung ca mt tam gic vung c di cnh huyn bng 5 2 /s: 14 2 m = HT 29. Cho hm s 3 2 22 ( 1) ( 4 3) 1. 3 y x m x m m x= + + + + + + Tm m hm s c cc tr. Tm gi tr ln nht ca biu thc 1 2 1 22( )A x x x x= + vi 1 2,x x l cc im cc tr ca hm s. 22. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 21 /s: 9 2 A khi 4m = HT 30. Cho hm s 3 2 3( 1) 9 (1)y x m x x m= + + vi m l tham s thc. Xc nh m hm s (1) t cc i , cc tiu sao cho 2CD CTy y+ = /s: 1 3 m m = = HT 31. Cho hm s (C3 2 21 ( 1) 1 ). 3 my x mx m x= + + Tm m hm s c cc i cc tiu v: D 2C CTy y+ > /s: 1 0 1 m m < < > HT 32. Cho hm s 3 2 3 2y x x= + (1). Tm im M thuc ng thng : 3 2d y x= sao tng khong cch t M ti hai im cc tr nh nht. /s: 4 2 ; 5 5 M HT 33. Cho hm s 3 2 2 3 3 3( 1)y x mx m x m m= + + (1). Tm m hm s (1) c cc tr ng thi khong cch t im cc i ca th hm s n gc ta O bng 2 ln khong cch t im cc tiu ca th hm s n gc ta O. /s: 3 2 2 3 2 2 m m = + = . HT 34. Cho hm s 3 2 3( 1) 3 ( 2) 2 ( )y x m x m m x m C= + + + + .Tm m th hm s (C) c cc tr ng thi khong cch t im cc i ca th hm s (C) ti trc Ox bng khong cch t im cc tiu ca th hm s (C) ti trc .Oy /s: 2; 1; 1; 0m m m m= = = = HT 35. Cho hm s 3 2 3 2y x x mx= + c th l (Cm). Tm m (Cm) c cc im cc i, cc tiu v ng thng i qua cc im cc tr song song vi ng thng : 4 3d y x= + ./s: 3m = HT 36. Cho hm s 3 2 3 2y x x mx= + c th l (Cm). Tm m (Cm) c cc im cc i, cc tiu v ng thng i qua cc im cc tr to vi ng thng : 4 5 0d x y+ = mt gc 0 45 . /s: 1 2 m = HT 37. Cho hm s 3 2 3y x x m= + + (1).Xc nh m th ca hm s (1) c hai im cc tr A, B sao cho 0 120AOB = . /s: 12 2 3 3 m + = HT 38. Cho hm s 3 2 2 3 3 3( 1) 4 1 (1),y x mx m x m m m= + + l tham s thc. Tm cc gi tr ca m th hm s (1) c hai im cc tr ,A B sao cho tam gic OAB vung ti ,O vi O l gc ta . /s: 1; 2m m= = HT 39. Cho hm s 3 2 3 2 3( 1) 3 ( 2) 3 .y x m x m m x m m= + + + + + + Chng minh rng vi mi m hm s lun c 2 cc tr v khong cch gia hai im ny khng ph thuc vo v tr ca m. HT 40. Cho hm s 3 2 3 2y x x mx= + (1) vi m l tham s thc. nh m hm s (1) c cc tr, ng thi ng thng i qua hai im cc tr ca th hm s to vi hai trc ta mt tam gic cn. /s: 3 2 m = HT 41. Cho hm s 4 2 2 ( ) 2( 2) 5 5y f x x m x m m= = + + + ( )mC . Tm cc gi tr ca m th ( )mC ca hm s c cc im cc i, cc tiu to thnh 1 tam gic vung cn. /s: 1m = HT 42. Cho hm s ( )4 2 2 2( 2) 5 5 .my x m x m m C= + + + Vi nhng gi tr no ca m th th (Cm) c im cc i v im cc tiu, ng thi cc im cc i v im cc tiu lp thnh mt tam gic u. /s: www.VNMATH.com 23. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 22 3 2 3m = . HT 43. Cho hm s 4 2 2 2y x mx m m= + + + c th (Cm) . Vi nhng gi tr no ca m th th (Cm) c ba im cc tr, ng thi ba im cc tr lp thnh mt tam gic c mt gc bng 0 120 . /s: 3 1 3 m = . HT 44. Cho hm s 4 2 2 1y x mx m= + c th (Cm) . Vi nhng gi tr no ca m th th (Cm) c ba im cc tr, ng thi ba im cc tr lp thnh mt tam gic c bn knh ng trn ngoi tip bng 1 . /s: 5 1 1; 2 m m = = HT 45. Cho hm s 4 2 4 2 2y x mx m m= + + c th (Cm) . Vi nhng gi tr no ca m th th (Cm) c ba im cc tr, ng thi ba im cc tr lp thnh mt tam gic c din tch bng 4. /s: 5 16m = . HT 46. Cho hm s 4 2 2 2x mx + c th ( )mC . Tm tt c cc gi tr ca tham s m th ( )mC c ba im cc tr to thnh mt tam gic c ng trn ngoi tip i qua im D 3 9 ; 5 5 /s: m = 1 PHN 3: S TNG GIAO HT 47. Cho hm s 3 2 6 9 6y x x x= + c th l (C). nh m ng thng ( ) : 2 4d y mx m= ct th (C) ti ba im phn bit. /s: 3m > HT 48. Cho hm s 3 2 3 2y x m x m= (Cm). Tm m (Cm) v trc honh c ng 2 im chung phn bit. /s: 1m = HT 49. Cho hm s 3 2 2 6 1y x x= + + . Tm m ng thng 1y mx= + ct (C) ti 3 im phn bit A, B, C sao cho A(0;1) v B l trung im ca AC. /s:m = 4 HT 50. Cho hm s 3 21 2 3 3 y x mx x m= + + co ox thi( )mC . Tmm e| ( )mC ca}ttrchoanhtai3ie|mphanbietco to|ng bnhphngcachoanhol nhn15. /s: 1m > HT 51. Cho hm s: 3 2 2 3 1 (1)y x x= + . Tm trn (C) nhng im M sao cho tip tuyn ca (C) ti M ct trc tung ti im c tung bng 8. /s: ( 1; 4)M HT 52. Cho hm s 3 2 2 ( 3) 4y x mx m x= + + + + c th l (Cm) (m l tham s).Cho ng thng (d): 4y x= + v im K(1; 3). Tm cc gi tr ca m (d) ct (Cm) ti ba im phn bit A(0; 4), B, C sao cho tam gic KBC c din tch bng 8 2 . /s: 1 137 2 m = . HT 53. Cho hm s 3 2 3 4y x x= + c th l (C). Gi kd l ng thng i qua im ( 1;0)A vi h s gc k ( )k . Tm k ng thng kd ct th (C) ti ba im phn bit A, B, C v 2 giao im B, C cng vi gc to O to thnh mt tam gic c din tch bng 1 . /s: 1k = HT 54. Cho hm s 3 2 3 2y x x= + c th l (C). Gi E l tm i xng ca th (C). Vit phng trnh ng thng qua E v ct (C) ti ba im E, A, B phn bit sao cho din tch tam gic OAB bng 2 . /s: ( )1; 1 3 ( 1)y x y x= + = . HT 55. Cho hm s 3 24 1 (2 1) ( 2) 3 3 y x m x m x= + + + + c th ( ),mC m l tham s. Gi A l giao im ca 24. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 23 ( )mC vi trc tung. Tm m sao cho tip tuyn ca ( )mC ti A to vi hai trc ta mt tam gic c din tch bng 1 . 3 /s: 13 11 ; 6 6 m m= = HT 56. Cho hm s 3 2y x mx= + + c th (Cm). Tm m th (Cm) ct trc honh ti mt im duy nht. /s: 3m > . HT 57. Cho hm s 3 2 2 3( 1) 6 2y x m x mx= + + c th (Cm).Tm m th (Cm) ct trc honh ti mt im duy nht. /s: 1 3 1 3m < < + HT 58. Cho hm s 3 2 3 1y x x= + . Tm m ng thng (): (2 1) 4 1y m x m= ct th (C) ti ng hai im phn bit. /s: 5 8 m = ; 1 2 m = . HT 59. Cho hm s 3 2 3 ( 1) 1y x mx m x m= + + + c th l ( )mC . Tm tt c cc gi tr ca m : 2 1d y x m= ct th ( )mC ti ba im phn bit c honh ln hn hoc bng 1. /s: khng c gi tr m HT 60. Cho hm s 3 3 2y x x= + (C). Vit phng trnh ng thng ct th (C) ti 3 im phn bit A, B, C sao cho 2Ax = v 2 2BC = /s: : 2d y x= + HT 61. Cho hm s 3 2 4 6 1y x mx= + (C), m l tham s. Tm m ng thng : 1d y x= + ct th hm s ti 3 im A(0;1), B, C vi B, C i xng nhau qua ng phn gic th nht. /s: 2 3 m = HT 62. Cho hm s 3 2 3 1y x x mx= + + + (m l tham s) (1).Tm m ng thng : 1d y = ct th hm s (1) ti ba im phn bit A(0; 1), B, C sao cho cc tip tuyn ca th hm s (1) ti B v C vung gc vi nhau. /s: 9 65 9 65 8 8 m m + = = HT 63. Cho hm s 3 3 1y x x= + c th (C) v ng thng (d): 3y mx m= + + . Tm m (d) ct (C) ti (1; 3)M , N, P sao cho tip tuyn ca (C) ti N v P vung gc vi nhau. /s: 3 2 2 3 2 2 3 3 m m + = = HT 64. Cho hm s 3 2 3 4y x x= + (C). Gi (d) l ng thng i qua im A(2; 0) c h s gc k. Tm k (d) ct (C) ti ba im phn bit A, M, N sao cho hai tip tuyn ca (C) ti M v N vung gc vi nhau. /s: 3 2 2 3 k = HT 65. Cho hm s 3 1 ( ).my x mx m C= + Tm m tip tuyn ca th hm s cho ti im 1x = ct ng trn (C): 2 2 ( 2) ( 3) 4x y + = theo mt dy cung c di nh nht. /s: 2m = HT 66. Cho hm s ( )3 3 2 .my x mx C= + Tm m ng thng i qua im cc i, cc tiu ca( )mC ct ng trn tm ( )1;1 ,I bn knh bng 1 ti hai im phn bit A, B sao cho din tch tam gic IAB t gi tr ln nht /s: 2 3 2 m = HT 67. Cho hm s 4 2 1y x mx m= + c th l ( )mC nh m th ( )mC ct trc trc honh ti bn im www.VNMATH.com 25. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 24 phn bit. /s: 1 2 m m > HT 68. Cho hm s 4 2 2( 1) 2 1 ( ).my x m x m C= + + + Tm tt c cc gi tr ca tham s m th hm s cho ct trc honh ti 4 im phn bit , , ,A B C D ln lt c honh 1 2 3 4, , ,x x x x 1 2 3 4( )x x x x< < < sao cho tam gic ACK c din tch bng 4 bit (3; 2).K /s: 4m = HT 69. Cho hm s ( )4 2 2 1 2 1y x m x m= + + + c th l ( )mC . nh m th ( )mC ct trc honh ti 4 im phn bit c honh lp thnh cp s cng. /s: 4 4; 9 m = HT 70. Cho hm s 4 2 (3 2) 3y x m x m= + + c th l (Cm), m l tham s. Tm m ng thng 1y = ct th (Cm) ti 4 im phn bit u c honh nh hn 2. /s: 1 1 3 0 m m < < HT 71. Cho hm s ( )4 2 2 1 2 1y x m x m= + + + c th l (Cm), m l tham s. Tm m th (Cm) ct trc honh ti 3 im phn bit u c honh nh hn 3. /s: 1 1 2 m m= . HT 72. Cho hm s: 4 2 5 4y x x= + . Tm tt c cc im M trn th (C) ca hm s sao cho tip tuyn ca (C) ti M ct (C) ti hai im phn bit khc M. /s: 10 10 2 2 30 6 m m < < HT 73. Cho hm s 2 1 2 x y x + = + c th l (C). Chng minh rng ng thng :d y x m= + lun ct th (C) ti hai im phn bit A, B. Tm m on AB c di nh nht. /s: 0m = . HT 74. Cho hm s 3 1 x y x = + (C). Vit phng trnh ng thng d qua im ( 1;1)I v ct th (C) ti hai im M, N sao cho I l trung im ca on MN. /s: 1y kx k= + + vi 0k < . HT 75. Cho hm s 2 4 1 x y x + = (C). Gi (d) l ng thng qua A(1; 1) v c h s gc k. Tm k (d) ct (C) ti hai im M, N sao cho 3 10MN = . /s: 3 41 3 41 3; ; 16 16 k k k + = = = HT 76. Cho hm s 2 2 1 x y x = + (C). Tm m ng thng (d): 2y x m= + ct (C) ti hai im phn bit A, B sao cho 5AB = ./s: 10; 2m m= = . HT 77. Cho hm s 1x y x m = + (1). Tm cc gi tr ca tham s m sao cho ng thng (d): 2y x= + ct th hm s (1) ti hai im A v B sao cho 2 2AB = . /s: 7m = HT 78. Cho hm s 2 ( ). 2 2 x y C x + = Tm tt c cc gi tr ca tham s m ng thng :d y x m= + ct th (C) ti hai im phn bit ,A B sao cho 2 2 37 2 OA OB+ = /s: 5 2 2 m m= = 26. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 25 HT 79. Cho hm s ( ). 1 x y C x = Tm tt c cc gi tr ca tham s m ng thng : 1d y mx m= ct th (C) ti hai im phn bit ,A B sao cho 2 2 MA MB+ t gi tr nh nht./s: 1m = HT 80. Cho hm s 1 ( ). 2 x y C x + = Gi d l ng thng qua (2; 0)M v c h s gc l k . Tm k d ct (C) ti hai im phn bit ,A B sao cho : 2MA MB= /s: 2 3 k = HT 81. Cho hm s 3 2 x y x + = + c th (H). Tm m ng thng d :y = 2x + 3m ct (H) ti hai im phn bit sao cho . 4OAOB = vi O l gc ta . /s: 7 12 m = HT 82. Tm trn (H) : 1 2 x y x + = cc im A, B sao cho di on thng AB bng 4 v ng thng AB vung gc vi ng thng .y x= /s: (3 2; 2); (3 2; 2) (3 2; 2); (3 2; 2)+ + A B hoac A B (1 2; 2 2); (1 2; 2 2) (1 2; 2 2); (1 2; 2 2)+ + + + A B hoac A B HT 83. Cho hm s 3 2 x y x + = c th (H). Tm m ng thng : 1d y x m= + + ti hai im phn bit A, B sao cho AOB nhn./s: 3m > HT 84. Cho hm s 3 2 ( ) 2 x y C x + = + . ng thng y x= ct (C) ti hai im A, B. Tm m ng thng y x m= + ct (C) ti hai im C, D sao cho ABCD l hnh bnh hnh. /s: 10m = HT 85. Cho hm s 2 1 1 x y x = (C). Tm m ng thng d: y x m= + ct (C) ti hai im phn bit A, B sao cho OAB vung ti O. /s: 2m = HT 86. Cho hm s 2 1 x m y mx = + (1). Chng minh rng vi mi 0m th hm s (1) ct (d) : 2 2y x m= ti hai im phn bit A, B thuc mt ng (H) c nh. ng thng (d) ct trc Ox, Oy ln lt ti cc im M, N. Tm m 3OAB OMNS S= HT 87. Cho hm s 2 1 ( ). 1 x y C x = Gi I l giao im ca hai tim cn ca (C). Vi gi tr no ca m th ng thng y x m= + ct th (C) ti hai im phn bit A, B v tam gic IAB u. /s: 3 6m = HT 88. Cho hm s ( ) 1 x y C x = . Tm cc gi tr ca m ng thng y x m= + ct th (C) ti hai im phn bit ,A B sao cho ,OA OB bng 0 60 . Vi O l gc ta . /s: 2 6m m= = PHN 4: TIP TUYN HT 89. Cho ham so 2 1 1 x y x = . Vit phng trnh tip tuyn ca (C), bit khong cch t im (1;2)I n tip tuyn bng 2 . /s: 1 0x y+ = v 5 0x y+ = www.VNMATH.com 27. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 26 HT 90. Cho hm s 3 2 (1 2 ) (2 ) 2y x m x m x m= + + + + (1) (m l tham s).Tm tham s m th ca hm s (1) c tip tuyn to vi ng thng d: 7 0x y+ + = gc , bit 1 cos 26 = ./s: 1 4 m hoc 1 2 m HT 91. Cho hm s 3 2 2 ( ).y x x x C= + Tm ta cc im trn trc honh sao cho qua im k c hai tip tuyn vi th (C) v gc gia hai tip tuyn ny bng 0 45 . /s: ;M O 32 ;0 27 M HT 92. Cho hm s 3 2 3 1y x x= + c th (C). Tm hai im A, B thuc th (C) sao cho tip tuyn ca (C) ti A v B song song vi nhau v di on AB = 4 2 . /s: (3;1), ( 1; 3)A B . HT 93. Cho hm s 1 1 x y x + = (C). Tm trn Oy tt c cc im t k c duy nht mt tip tuyn ti (C). /s: (0;1); (0; 1)M M HT 94. Cho hm s 3 3y x x= (C). Tm trn ng thng :d y x= cc im m t k c ng 2 tip tuyn phn bit vi th (C)./s: (2; 2); ( 2;2)A B HT 95. Cho hm s: 3 3 2y x x= + . Tm tt c im trn ng thng 4y = , sao cho t k c ng 2 tip tuyn ti th (C). /s: 2 ( 1;4); ;4 ;(2;4) 3 HT 96. Cho hm s 3 2 3 2y x x= + (C). Tm trn ng thng : 2d y = cc im m t k c 3 tip tuyn phn bit vi th (C). /s: 1 5 3 2 m m m < > HT 97. Cho hm s ( ) ( ) 2 2 1 . 1y x x= + (C). Cho im ( ;0)A a . Tm a t A k c 3 tip tuyn phn bit vi th (C). /s: 3 3 1 1 2 2 < >a hoac a HT 98. Cho hm s 3 2 3 2.y x x= + Tm trn ng thng 2y = cc im m t c th k c 2 tip tuyn ti th hm s v 2 tip tuyn vung gc vi nhau. /s: 1 2; 27 M HT 99. Cho hm s 3 21 ( ) ( 1) (4 3 ) 1 3 y f x mx m x m x= = + + + c th l (Cm). Tm cc gi tr m sao cho trn th (Cm) tn ti mt im duy nht c honh m m tip tuyn ti vung gc vi ng thng : 2 3 0d x y+ = . /s: hay 2 0 3 m m< > . HT 100. Tm tt c cc gi tr m sao cho trn th ( )mC : 3 21 ( 1) (4 3) 1 3 y mx m x m x= + + + tn ti ng hai im c honh dng m tip tuyn ti vung gc vi ng thng (L): 2 3 0x y+ = /s: 1 1 2 0; ; 2 2 3 m HT 101. Cho hm s 2 2 x y x = + (C). Vit phng trnh tip tuyn ca th (C), bit rng khong cch t tm i xng ca th (C) n tip tuyn l ln nht. /s:y x= v 8y x= + . HT 102. Cho hm s 2 2 3 x y x + = + (1). Vit phng trnh tip tuyn ca th hm s (1), bit tip tuyn ct 28. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 27 trc honh, trc tung ln lt ti hai im phn bit A, B v tam gic OAB cn ti gc ta O. /s: 2y x= . HT 103. Cho hm s y = 2 1 1 x x . Lp phng trnh tip tuyn ca th (C) sao cho tip tuyn ny ct cc trc Ox, Oy ln lt ti cc im A v B tho mn OA = 4OB. /s: 1 5 4 4 1 13 4 4 y x y x = + = + . HT 104. Vit phng trnh tip tuyn ca th hm s 2 2 x y x = bit tip tuyn ct Ox, Oy ln lt ti A v B m tam gic OAB tha mn: 2AB OA= /s: 8y x= + HT 105.Cho hm s 2 3 2 x y x = c th (C). Tm trn (C) nhng im M sao cho tip tuyn ti M ca (C) ct hai tim cn ca (C) ti A, B sao cho AB ngn nht. /s: (3;3)M hoc (1;1)M HT 106.Cho hm s 2 3 2 x y x = .Cho M l im bt k trn (C). Tip tuyn ca (C) ti M ct cc ng tim cn ca (C) ti A v B. Gi I l giao im ca cc ng tim cn. Tm to im M sao cho ng trn ngoi tip tam gic IAB c din tch nh nht. /s: (1;1); (3;3)M M HT 107.Cho hm s 3 2 2 2 1 ( ).my x mx m x m C= + + Tm m th hm s tip xc vi trc honh. /s: 3 1 3 2 m m m= = = HT 108.Cho hm s 2 1 1 x y x + = c th (C). Gi I l giao im ca hai tim cn. Tm im M thuc (C) sao cho tip tuyn ca (C) ti M ct 2 tim cn ti A v B vi chu vi tam gic IAB t gi tr nh nht. /s: ( )1 1 3;2 3M + + , ( )2 1 3;2 3M HT 109. Cho hm s: 2 1 x y x + = (C). Cho im (0; )A a . Tm a t A k c 2 tip tuyn ti th (C) sao cho 2 tip im tng ng nm v 2 pha ca trc honh. /s: 2 3 1 a a > . HT 110. Cho hm s y = 2 1 x x + + . Gi I l giao im ca 2 ng tim cn, l mt tip tuyn bt k ca th (C). d l khong cch t I n . Tm gi tr ln nht ca d. /s:GTLN ca d bng 2 khi 0 0 0 2 x x = = HT 111. Cho hm s 2 1 1 x y x + = + . Vit phng trnh tip tuyn ca th (C), bit rng tip tuyn cch u hai im A(2; 4), B(4; 2). /s: 1 5 ; 1; 5 4 4 y x y x y x= + = + = + HT 112. Cho hm s 2 3 2 x y x = (C). Vit phng trnh tip tuyn ti im M thuc (C) bit tip tuyn ct tim cn ng v tim cn ngang ln lt ti A, B sao cho csin gc ABI bng 4 17 , vi I l giao 2 tim cn. /s: Ti 3 0; 2 M : 1 3 4 2 y x= + ; Ti 5 4; 3 M : 1 7 4 2 y x= + www.VNMATH.com 29. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 28 HT 113. Cho hm s 1 2 1 x y x + = (C). Tm gi tr nh nht ca m sao cho tn ti t nht mt im M (C) m tip tuyn ti M ca (C) to vi hai trc ta mt tam gic c trng tm nm trn ng thng 2 1y m= /s: 1 3 m HT 114. Cho hm s 2 1 ( ). 1 x y C x = Tm cc gi tr ca m th hm s (C) tip xc vi ng thng 5.y mx= + /s: 1m = hoc 9m = PHN 5: BIN LUN S NGHIM CA PHNG TRNH HT 115. Cho hm s 3 2 3 1y x x= + + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm m phng trnh 3 2 3 2 3 3x x m m = c ba nghim phn bit. /s: {( 1;3) 0;2}m HT 116.Cho hm s 4 2 5 4y x x= + c th (C). 1) Kho st s bin thin v v th (C) ca hm s. 2) Tm m phng trnh 4 2 2| 5 4 | logx x m + = c 6 nghim. Da vo th ta c PT c 6 nghim 9 44 12 9 log 12 144 12 4 m m= = = . HT 117. Cho hm s: 4 2 2 1y x x= + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Bin lun theo m s nghim ca phng trnh: 4 2 22 1 log 0x x m + + = (m> 0) 1 0 2 m< < 1 2 m = 1 1 2 m< < 1m = 1m > 2 nghim 3 nghim 4 nghim 2 nghim v nghim HT 118. Cho hm s 4 2 ( ) 8 9 1y f x x x= = + . 1) Kho st s bin thin v v th (C) ca hm s. 2) Da vo th (C) hy bin lun theo m s nghim ca phng trnh: 4 2 8 cos 9 cos 0x x m + = vi [0; ]x /s: 0m < 0m = 0 1m< < 81 1 32 m < 81 32 m = 81 32 m > 30. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 29 v nghim 1 nghim 2 nghim 4 nghim 2 nghim v nghim HT 119. Cho hm s 1 . 1 x y x + = 1) Kho st s bin thin v v th (C) ca hm s. 2) Bin lun theo m s nghim ca phng trnh 1 . 1 x m x + = 1; 1m m< > 1m = 1 1m < 2 nghim 1 nghim v nghim PHN 6: IM C BIT CA TH HT 120. Cho hm s 3 3 2y x x= + + (C). Tm 2 im trn th hm s sao cho chng i xng nhau qua tm ( 1;3)M . /s:( )1;0 v ( )1;6 HT 121. Cho hm s 3 3 2y x x= + + (C). Tm trn (C) hai im i xng nhau qua ng thng : 2 2 0d x y + = . /s: 7 1 7 7 1 7 ;2 ; ;2 2 2 2 2 2 2 + HT 122. Cho hm s x 3 2 11 3 3 3 x y x= + + . Tm trn th (C) hai im phn bit M, N i xng nhau qua trc tung. /s: 16 16 3; , 3; 3 3 M N . HT 123. Cho hm s 2 1 1 x y x = + (C).Tm im M thuc th (C) tip tuyn ca (C) ti M vi ng thng i qua M v giao im hai ng tim cn c tch cc h s gc bng 9. /s: (0; 3); ( 2;5)M M HT 124. Cho hm s 2 1 1 x y x + = + (C). Tm trn (C) nhng im c tng khong cch n hai tim cn ca (C) nh nht. /s:(0;1);( 2;3) HT 125. Cho hm s 3 4 2 x y x = (C). Tm cc im thuc (C) cch u 2 tim cn. /s: 1 2(1;1); (4;6)M M HT 126. Cho hm s 4 21 1 1 ( ). 4 2 y x x C= + Tm im M thuc (C) sao cho tng khong cch t im M n hai trc ta l nh nht. /s: (0;1)M HT 127. Cho hm s 4 2 0 0 0 02 3 2 1y x x x= + + c th l (C) v ng thng ( ) 2 1x = .Tm trn th (C) im A c khong cch n ( ) l nh nht /s: 1 3 1 ; 3 2 8 A 2 3 1 ; ; 3 2 8 A + www.VNMATH.com 31. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 30 HT 128. Cho hm s 1 2 x y x + = . Tm trn th hm s im M sao cho tng khong cch t M n hai trc ta l nh nht. /s: 1 0; 2 M HT 129. Cho hm s 2 4 1 x y x = + . Tm trn (C) hai im i xng nhau qua ng thng MN bit ( 3;0); ( 1; 1)M N /s:A(0; 4), B(2; 0). HT 130. Cho hm s 2 1 x y x = . Tm trn th (C) hai im B, C thuc hai nhnh sao cho tam gic ABC vung cn ti nh A vi A(2; 0). /s: ( 1;1), (3;3)B C HT 131. Cho hm s 2 1 1 x y x = + . Tm ta im M (C) sao cho khong cch t im ( 1; 2)I ti tip tuyn ca (C) ti M l ln nht. /s: ( )1 3;2 3M + hoc ( )1 3;2 3M + HT 132. Cho hm s 2 2 1 x y x + = . Tm nhng im trn th (C) cch u hai im (2;0), (0;2)A B . /s: 1 5 1 5 1 5 1 5 , ; , 2 2 2 2 + + HT 133. Cho hm s 3 1 x y x = + . Tm trn hai nhnh ca th (C) hai im A v B sao cho AB ngn nht./s: ( ) ( )4 4 4 4 1 4;1 64 , 1 4;1 64A B + + . HT 134. Cho hm s 4 2 2 1y x x= + Tm ta hai im P. Q thuc (C) sao cho ng thng PQ song song vi trc honh v khong cch t im cc i ca (C) n ng thng PQ bng 8 /s:Vy, P(-2;9), Q(2;9) hoc P(2;9); Q(-2;9) HT 135. Cho hm s 2 (3 1) . m x m m y x m + + = + Tm cc im thuc ng thng 1x = m khng c th i qua. /s:Tp hp cc im thuc ng thng 1x = c tung bng a vi a tha mn : 2 10a< < HT 136. Cho hm s 2 1 ( ). 1 x y C x = Tm trn th (C) hai im ,A B phn bit sao cho ba im , , (0; 1)A B I thng hng ng thi tha mn: . 4.IAIB = /s: ( ) ( )2 2;1 2 ; 2 2;1 2A B + + hoc ( ) ( )1 3; 2 3 ; 1 3; 2 3A B + + PHN 7: CC BI TNG HP HT 137.Cho hm s 2 3 ( ). 2 x y C x + = Tm m ng thng : 2d y x m= + ct th ti hai im phn bit sao cho tip tuyn ti hai im ca th hm s song song vi nhau. /s: 2m = HT 138.Cho hm s 3 2 2 2 1 ( ).y x mx mx C= + Tm m th hm s (C) ct trc honh ti 3 im phn bit (1;0),A B v C sao cho 1 2 . 5k k BC+ = trong 1 2,k k ln lt l h s gc tip tuyn ti B, C ca th hm s (C). /s: 1; 2m m= = HT 139.Cho hm s 3 2 3 2 ( ).my x x mx m C= + + Tm m ( )mC ct trc honh ti 3 im phn bit , ,A B C 32. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 31 sao cho tng cc h s gc ca tip tuyn ca ( )mC ti , ,A B C bng 3./s: 2m = www.VNMATH.com 33. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 32 PHN 8: TUYN TP THI I HC T NM 2009 HT 140.(H A 2009) Cho hm s 2 (1) 2 3 x y x + = + . Vit phng trnh tip tuyn ca th hm s (1), bit tip tuyn ct trc honh, trc tung ln lt ti hai im phn bit A, B sao cho tam gic OAB cn ti gc ta O. /s: 2y x= HT 141.(H B 2009) Cho hm s: 4 2 2 4 (1)y x x= .Vi gi tr no ca ,m phng trnh 2 2 2x x m = c ng 6 nghim thc phn bit. /s: 0 1m< < HT 142.(H D 2009) Cho hm s 4 2 (3 2) 3y x m x m= + + c th l ( )mC vi m l tham s. Tm m ng thng 1y = ct th ( )mC ti 4 im phn bit u c honh nh hn 2. /s: 1 1, 0 3 m m < < HT 143.(H A 2010) Cho hm s 3 2 2 (1 ) (1),y x x m x m= + + vi m l tham s thc. Tm m th (1) ct trc honh ti 3 im phn bit c honh 1 2 3, ,x x x tha mn iu kin: 2 2 2 1 2 3 4x x x+ + < /s: 1 1 4 m < < v 0m HT 144. (H B 2010) Cho hm s 2 1 ( ) 1 x y C x + = + . Tm m ng thng 2y x m= + ct th ( )C ti hai im A v B sao cho tam gic OAB c din tch bng 3 (O l gc ta ). /s: 2m = HT 145. (D 2010) Cho hm s 4 2 6 ( )y x x C= + . Vit phng trnh tip tuyn ca th (C) bit tip tuyn vung gc vi ng thng 1 1 6 y x= /s: 6 10y x= + HT 146. (A 2011)Cho hm s 1 ( ) 2 1 x y C x + = . Chng minh rng vi mi m ng thng y x m= + lun ct th ( )C ti hai im phn bit A v B. Gi 1 2,k k ln lt l h s gc ca tip tuyn vi ( )C ti A v B. Tm m tng 1 2k k+ t gi tr ln nht. /s: 1 2k k+ ln nht bng 2 , khi v ch khi 1.m = HT 147. (B 2011) Cho hm s 4 2 2( 1) (1)y x m x m= + + (vi m l tham s). Tm m th hm s (1) c ba im cc tr A, B, C sao cho OA = BC; trong O l gc ta , A l im cc tr thuc trc tung, B v C l hai im cc tr cn li. /s: 2 2 2m = HT 148. (D 2011) Cho hm s 2 1 ( ) 1 x y C x + = + . Tm k ng thng 2 1y kx k= + + ct th ( )C ti hai im phn bit ,A B sao cho khong cch t A v B n trc honh bng nhau. /s: HT 149. (A,A1 2012) Cho hm s 4 2 2 2( 1) (1)y x m x m= + + , vi m l tham s thc. Tm m th hm s (1) c 3 im cc tr to thnh 3 nh ca mt tam gic vung. /s: 0m = HT 150. (B 2012) Cho hm s 3 2 3 3 3 (1),y x mx m= + m l tham s thc. Tm m th hm s (1) c hai im cc tr A v B sao cho tam gic OAB c din tch bng 48. /s: 2m = HT 151. (D 2012) Cho hm s 3 2 22 2 2(3 1) (1), 3 3 y x mx m x m= + l tham s thc. Tm m hm s (1) c hai im cc tr 1 2;x x sao cho: 1 2 1 22( ) 1.x x x x+ + = /s: 2 3 m = HT 152. (A,A1 2013) Cho hm s 3 2 3 3 1 (1)y x x mx= + + , vi m l tham s thc. Tm m hm s (1) 3k = 34. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 33 nghch bin trn khong (0; )+ /s: 1m HT 153. (B 2013) Cho hm s 3 2 2 3( 1) 6 (1),y x m x mx= + + vi m l tham s thc. Tm m th hm s (1) c hai im cc tr A, B sao cho ng thng AB vung gc vi ng thng 2.y x= + /s: 0; 2m m= = HT 154. (D 2013) Cho hm s 3 2 2 3 ( 1) 1 (1),y x mx m x= + + vi m l tham s thc. Tm m ng thng 1y x= + ct th hm s (1) ti ba im phn bit. /s: 8 0; 9 m m< > -----------------------------------------------------------HT----------------------------------------------------------- www.VNMATH.com 35. CHUYN LUYN THI I HC 2013 - 2014 PHNG TRNH M - LOGARIT BIN SON: LU HUY THNG H NI, 8/2013 H V TN: LP :. TRNG : 36. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 1 CHUYN : PHNG TRNH M LOGARIT VN I: LY THA 1. nh ngha lu tha S m C s a Lu tha a * n N = a R . ......n a a a a a = = (n tha s a) 0 = 0a 0 1a a = = * ( )n n N = 0a 1n n a a a = = * ( , ) m m Z n N n = 0a > ( ) m n nm nna a a a b b a = = = = * lim ( , )n n r r Q n N = 0a > lim n r a a = 2. Tnh cht ca lu tha Vi mi a > 0, b > 0 ta c: . . ; ; ( ) ; ( ) . ; a a a a a a a a a ab a b ba b + = = = = = a > 1 : a a > > ; 0 < a < 1 : a a > < Vi 0 < a < b ta c: 0m m a b m< > ; 0m m a b m> < Ch : + Khi xt lu tha vi s m 0 v s m nguyn m th c s a phi khc 0. + Khi xt lu tha vi s m khng nguyn th c s a phi dng. 3. nh ngha v tnh cht ca cn thc Cn bc n ca a l s b sao cho n b a= . Vi a, b 0, m, n N*, p, q Z ta c: .n n n ab a b= ; ( 0) n n n a a b b b = > ; ( ) ( 0) p n np a a a= > ; m n mn a a= ( 0) n mp qp q Neu th a a a n m = = > ; c bit mnn m a a= www.VNMATH.com 37. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 2 Nu n l s nguyn dng l v a < b th n n a b< . Nu n l s nguyn dng chn v 0 < a < b th n n a b< . Ch : + Khi n l, mi s thc a ch c mt cn bc n. K hiu n a . + Khi n chn, mi s thc dng a c ng hai cn bc n l hai s i nhau. 4. Cng thc li kp Gi A l s tin gi, r l li sut mi k, N l s k. S tin thu c (c vn ln li) l: (1 )N C A r= + VN II: LOGARIT 1. nh ngha Vi a > 0, a 1, b > 0 ta c: loga b a b = = Ch : loga b c ngha khi 0, 1 0 a a b > > Logarit thp phn: 10 lg log logb b b= = Logarit t nhin (logarit Nepe): ln loge b b= (vi 1 lim 1 2,718281 n e n = + ) 2. Tnh cht log 1 0a = ; log 1a a = ; log b a a b= ; log ( 0)a b a b b= > Cho a > 0, a 1, b, c > 0. Khi : + Nu a > 1 th log loga a b c b c> > + Nu 0 < a < 1 th log loga a b c b c> < 3. Cc qui tc tnh logarit Vi a > 0, a 1, b, c > 0, ta c: log ( ) log loga a a bc b c= + log log loga a a b b c c = log loga a b b = 4. i c s 38. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 3 Vi a, b, c > 0 v a, b 1, ta c: log log log a b a c c b = hay log .log loga b a b c c= 1 log loga b b a = 1 log log ( 0)aa c c = Bi tp c bn HT 1: Thc hin cc php tnh sau: 1) 2 1 4 log 4.log 2 2) 5 27 1 log .log 9 25 3) 3 loga a 4) 32 log 2log 3 4 9+ 5) 2 2 log 8 6) 9 8 log 2 log 27 27 4+ 7) 3 4 1/3 7 1 log .log log a a a a a a 8) 3 8 6 log 6.log 9.log 2 9) 3 81 2 log 2 4 log 5 9 + 10) 3 9 9 log 5 log 36 4 log 7 81 27 3+ + 11) 75 log 8log 6 25 49+ 12) 2 5 3 log 4 5 13) 6 8 1 1 log 3 log 2 9 4+ 14) 9 2 125 1 log 4 2 log 3 log 27 3 4 5 + + + 15) 36 log 3.log 36 HT 2: So snh cc cp s sau: 1) 4 va log3 1 log 4 3 2) 0,2 va log3 0,1 log 2 0,34 3) 5 2 va log3 4 2 3 log 5 4 4) 1 1 3 2 1 1 log log 80 15 2 va + 5) 13 17 log 150 log 290va 6) va 6 6 1 loglog 3 22 3 HT 3: Tnh gi tr ca biu thc logarit theo cc biu thc cho: 1)Cho 2 log 14 a= . Tnh 49 log 32 theo a. 2)Cho 15 log 3 a= . Tnh 25 log 15 theo a. 3)Cho lg3 0,477= . Tnh lg9000; lg0,000027 ; 81 1 log 100 . 4)Cho 7 log 2 a= . Tnh 1 2 log 28 theo a. HT 4: Tnh gi tr ca biu thc logarit theo cc biu thc cho: 1)Cho 25 log 7 a= ; 2 log 5 b= . Tnh 3 5 49 log 8 theo a, b. www.VNMATH.com 39. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 4 2)Cho 30 log 3 a= ; 30 log 5 b= . Tnh 30 log 1350 theo a, b. 3)Cho 14 log 7 a= ; 14 log 5 b= . Tnh 35 log 28 theo a, b. 4)Cho 2 log 3 a= ; 3 log 5 b= ; 7 log 2 c= . Tnh 140 log 63 theo a, b, c. VN III: HM S LY THA HM S M HM S LOGARIT 1. Khi nim 1)Hm s lu tha y x = ( l hng s) S m Hm s y x = Tp xc nh D = n (n nguyn dng) n y x= D = R = n (n nguyn m hoc n = 0) n y x= D = R{0} l s thc khng nguyn y x = D = (0; +) Ch : Hm s 1 ny x= khng ng nht vi hm s ( *)n y x n N= . 2)Hm s m x y a= (a > 0, a 1). Tp xc nh: D = R. Tp gi tr: T = (0; +). Khi a > 1 hm s ng bin, khi 0 < a < 1 hm s nghch bin. Nhn trc honh lm tim cn ngang. th: 01 y=ax y x1 40. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 5 3)Hm s logarit loga y x= (a > 0, a 1) Tp xc nh: D = (0; +). Tp gi tr: T = R. Khi a > 1 hm s ng bin, khi 0 < a < 1 hm s nghch bin. Nhn trc tung lm tim cn ng. th: 2. Gii hn c bit 1 0 1 lim(1 ) lim 1 x x x x x e x + = + = 0 ln(1 ) lim 1 x x x + = 0 1 lim 1 x x e x = 3. o hm ( ) 1 ( 0)x x x = > ; ( ) 1 .u u u = Ch : ( ) 1 01 0 > = n n n vi x neu n chan x vi x neu n len x . ( ) 1 n n n u u n u = ( ) lnx x a a a = ; ( ) ln .u u a a a u = ( )x x e e = ; ( ) .u u e e u = ( ) 1 log lna x x a = ; ( )log lna u u u a = ( ) 1 ln x x = (x > 0); ( )ln u u u = 01 y=logax 1 y x O www.VNMATH.com 41. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 6 42. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 7 Bi tp c bn HT 5: Tnh cc gii hn sau: 1) lim 1 x x x x+ + 2) 1 1 lim 1 x x x x + + + 3) 2 1 1 lim 2 x x x x + + 4) 1 33 4 lim 3 2 x x x x + + + 5) 1 lim 2 1 x x x x+ + 6) 2 1 lim 1 x x x x+ + 7) ln 1 lim x e x x e 8) 2 0 1 lim 3 x x e x i) 1 lim 1 x x e e x k) 0 lim sin x x x e e x l) sin 2 sin 0 lim x x x e e x m) ( )1 lim 1x x x e + HT 6: Tnh o hm ca cc hm s sau: 1) 3 2 1y x x= + + 2) 4 1 1 x y x + = 3) 2 5 2 2 1 x x y x + = + 4) 3 sin(2 1)y x= + 5) 3 2 cot 1y x= + 6) 3 3 1 2 1 2 x y x = + 7) 3 3 sin 4 x y + = 8) 11 5 9 9 6y x= + 9) 2 4 2 1 1 x x y x x + + = + HT 7: Tnh o hm ca cc hm s sau: 1) 2 ( 2 2) x y x x e= + 2) 2 ( 2 ) x y x x e = + 3) 2 .sinx y e x = 4) 2 2x x y e + = 5) 1 3. x x y x e = 6) 2 2 x x x x e e y e e + = 7) cos 2 .x x y e= 8) 2 3 1 x y x x = + i) cot cos . x y x e= HT 8: Tnh o hm ca cc hm s sau: 1) 2 ln(2 3)y x x= + + 2) 2 log (cos )y x= 3) .ln(cos )x y e x= 4) 2 (2 1)ln(3 )y x x x= + 5) 3 1 2 log ( cos )y x x= 6) 3 log (cos )y x= 7) ln(2 1) 2 1 x y x + = + 8) ln(2 1) 1 x y x + = + 9) ( )2 ln 1y x x= + + HT 9: Chng minh hm s cho tho mn h thc c ch ra: 1) 2 22. ; (1 ) x y x e xy x y = = 2) ( 1) ;x x y x e y y e= + = www.VNMATH.com 43. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 8 3) 4 2 ; 13 12 0x x y e e y y y = + = 4) 2 . . ; 3 2 0x x y a e be y y y = + + + = 5) .sin ; 2 2 0x y e x y y y = + + = 6) ( )4 .cos ; 4 0x y e x y y = + = HT 10: Chng minh hm s cho tho mn h thc c ch ra: 1) 1 ln ; 1 1 y y xy e x = + = + 2) 1 ; ln 1 1 ln y xy y y x x x = = + + 3) 2 sin(ln ) cos(ln ); 0y x x y xy x y= + + + = 4) 2 2 21 ln ; 2 ( 1) (1 ln ) x y x y x y x x + = = + HT 11: Gii phng trnh, bt phng trnh sau vi hm s c ch ra: 1) 2 '( ) 2 ( ); ( ) ( 3 1)x f x f x f x e x x= = + + 2) 31 '( ) ( ) 0; ( ) lnf x f x f x x x x + = = 3) 2 1 1 2 '( ) 0; ( ) 2. 7 5x x f x f x e e x = = + + VN IV: PHNG TRNH M 1. Phng trnh m c bn: Vi 0, 1> a a : 0 log x a b a b x b >= = 2. Mt s phng php gii phng trnh m 1) a v cng c s: Vi 0, 1> a a : ( ) ( ) ( ) ( )f x g x a a f x g x= = Ch : Trong trng hp c s c cha n s th: ( 1)( ) 0M N a a a M N= = 2) Logarit ho: ( )( ) ( ) ( ) log . ( )f x g x a a b f x b g x= = 3) t n ph: Dng 1: ( ) ( ) 0f x P a = ( ) , 0 ( ) 0 f x t a t P t = > = , trong P(t) l a thc theo t. Dng 2: 2 ( ) ( ) 2 ( ) ( ) 0f x f x f x a ab b + + = Chia 2 v cho 2 ( )f x b , ri t n ph ( )f x a t b = Dng 3: ( ) ( )f x f x a b m+ = , vi 1ab = . t ( ) ( ) 1f x f x t a b t = = 44. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 9 4) S dng tnh n iu ca hm s Xt phng trnh: f(x) = g(x) (1) on nhn x0 l mt nghim ca (1). Da vo tnh ng bin, nghch bin ca f(x) v g(x) kt lun x0 l nghim duy nht: ong bien va nghch bien (hoac ong bien nhng nghiem ngat). n ieu va hang so ( ) ( ) ( ) ( ) f x g x f x g x c = Nu f(x) ng bin (hoc nghch bin) th ( ) ( )f u f v u v= = 5) a v phng trnh cc phng trnh c bit Phng trnh tch A.B = 0 0 0 A B = = Phng trnh 2 2 0 0 0 A A B B =+ = = 6) Phng php i lp Xt phng trnh: f(x) = g(x) (1) Nu ta chng minh c: ( ) ( ) f x M g x M th (1) ( ) ( ) f x M g x M = = Bi tp c bn HT 12: Gii cc phng trnh sau (a v cng c s hoc logarit ho): 1) 3 1 8 2 9 3x x = 2) ( ) 2 3 2 2 3 2 2 x = + 3) 2 2 2 3 2 6 5 2 3 7 4 4 4 1x x x x x x + + + + + + = + 4) 2 2 5 7 5 .35 7 .35 0x x x x + = 5) 2 2 2 2 1 2 1 2 2 3 3x x x x + + = + 6) 2 4 5 25x x + = 7) 2 2 4 31 2 2 x x = 8) 7 1 2 1 1 . 2 2 2 x x+ = 9) 1 3 .2 72x x + = 10) 1 1 5 6. 5 3. 5 52x x x+ + = 11) 10 5 10 1516 0,125.8 x x x x + + = 12) ( ) ( ) 1 1 1 5 2 5 2 x x x + + = HT 13: Gii cc phng trnh sau (a v cng c s hoc logarit ho): 1) 4 1 3 2 2 1 5 7 x x+ + = 2) 2 1 15 .2 50 x x x + = 3) 3 23 .2 6 x x x + = 4) 23 .8 6 x x x+ = 5) 1 2 1 4.9 3 2x x + = 6) 2 2 2 .3 1,5x x x = www.VNMATH.com 45. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 10 7) 2 5 .3 1x x = 8) 3 2 2 3 x x = 9) 2 3 .2 1x x = HT 14: Gii cc phng trnh sau (t n ph dng 1): 1) 1 4 2 8 0x x + + = 2) 1 1 4 6.2 8 0x x+ + + = 3) 4 8 2 5 3 4.3 27 0x x+ + + = 4) 16 17.4 16 0x x + = 5) 1 49 7 8 0x x + + = 6) 2 2 2 2 2 3.x x x x + = 7) ( ) ( )7 4 3 2 3 6 x x + + + = 8) 2 cos2 cos 4 4 3x x + = 9) 2 5 1 3 36.3 9 0x x+ + + = 10) 2 2 2 2 1 3 28.3 9 0x x x x+ + + + = 11) 2 2 2 2 4 9.2 8 0x x+ + + = 12) 2 1 1 3.5 2.5 0,2x x = HT 15: Gii cc phng trnh sau (t n ph dng 1): 1) 25 2(3 ).5 2 7 0x x x x + = 2) 2 2 3.25 (3 10).5 3 0x x x x + + = 3) 3.4 (3 10).2 3 0x x x x+ + = 4) 9 2( 2).3 2 5 0x x x x+ + = 5) 2 1 2 4 .3 3 2.3 . 2 6x x x x x x x+ + + = + + 6) 2 2 3.25 (3 10).5 3 0x x x x + + = 7) 4 +( 8 2 +12 2 ) 0x x x x = 8) 4 9 5 3 1( ). ( ). 0x x x x+ + + = 9) 2 22 2 4 ( 7).2 12 4 0x x x x+ + = 10) 9 ( 2).3 2( 4) 0x x x x + + = HT 16: Gii cc phng trnh sau (t n ph dng 2): 1) 64.9 84.12 27.16 0x x x + = 2) 3.16 2.81 5.36x x x + = 3) 2 2 6.3 13.6 6.2 0x x x + = 4) 2 1 25 10 2x x x+ + = 5) 27 12 2.8x x x + = 6) 3.16 2.81 5.36x x x + = 7) 1 1 1 6.9 13.6 6.4 0x x x + = 8) 1 1 1 4 6 9x x x + = 9) 1 1 1 2.4 6 9x x x+ = 10) ( ) ( )( ) ( )7 5 2 2 5 3 2 2 3 1 2 1 2 0. x x x + + + + + + = HT 17: Gii cc phng trnh sau (t n ph dng 3): 1) ( ) ( )2 3 2 3 14 x x + + = 2) ( ) ( )2 3 2 3 4 x x + + = 3) (2 3) (7 4 3)(2 3) 4(2 3)x x + + + = + 4) ( ) ( ) 3 5 21 7 5 21 2 x x x+ + + = 5) ( ) ( )5 24 5 24 10 x x + + = 6) 7 3 5 7 3 5 7 8 2 2 x x + + = 7) ( ) ( )6 35 6 35 12 x x + + = 8) ( ) ( ) 2 2 ( 1) 2 1 4 2 3 2 3 2 3 x x x + + = 46. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 11 9) ( ) ( ) 3 3 5 16 3 5 2 x x x + + + = 10) ( ) ( )3 5 3 5 7.2 0 x x x + + = 11) ( ) ( )7 4 3 3 2 3 2 0 x x + + = 12) ( ) ( )3 3 3 8 3 8 6. x x + + = HT 18: Gii cc phng trnh sau (s dng tnh n iu): 1)( ) ( )2 3 2 3 4 x x x + + = 2) ( ) ( ) ( )3 2 3 2 10 x x x + + = 3) ( ) ( )3 2 2 3 2 2 6 x x x + + = 4) ( ) ( ) 3 3 5 16. 3 5 2 x x x+ + + = 5) 3 7 2 5 5 x x + = 6) ( ) ( )2 3 2 3 2 x x x + + = 7) 2 3 5 10x x x x + + = 8) 2 3 5x x x + = 9) 2 1 2 2 2 ( 1)x x x x = 10) 3 5 2x x= 11) 2 3x x= 12) 1 2 4 1x x x+ = HT 19: Gii cc phng trnh sau (a v phng trnh tch): 1) 8.3 3.2 24 6x x x + = + 2) 1 12.3 3.15 5 20x x x + + = 3) 3 8 .2 2 0x x x x + = 4) 2 3 1 6x x x + = + 5) 2 2 2 3 2 6 5 2. 3 7 4 4 4 1x x x x x x + + + + + + = + 6) ( ) 2 2 2 11 4 2 2 1 xx x x ++ + = + 7) 2 3 2 .3 3 (12 7 ) 8 19 12x x x x x x x+ = + + 8) 2 1 1 .3 (3 2 ) 2(2 3 )x x x x x x x + = 9) sin 1 sin 4 2 cos( ) 2 0yx x xy+ + = 10) 2 2 2 2 2( ) 1 2( ) 1 2 2 2 .2 1 0x x x x x x+ + + = HT 20: Gii cc phng trnh sau (phng php i lp): 1) 4 2 cos ,x x= vi x 0 2) 2 6 10 2 3 6 6x x x x + = + 3) sin 3 cosx x= 4) 3 2 2.cos 3 3 2 x xx x = + 5) sin cos x x = 6) 2 2 2 1 2 x x x x + = 7) 2 3 cos2x x= 8) 2 5 cos3x x= HT 21: Tm m cc phng trnh sau c nghim: 1) 9 3 0x x m+ + = 2) 9 3 1 0x x m+ = 3) 1 4 2x x m+ = 4) 2 3 2.3 ( 3).2 0x x x m+ + = 5) 2 ( 1).2 0x x m m + + + = 6) 25 2.5 2 0x x m = 7) 2 16 ( 1).2 1 0x x m m + = 8) 25 .5 1 2 0x x m m+ + = 9) 2 2 sin os 81 81x c x m+ = 10) 2 2 4 2 2 3 2.3 2 3 0x x m + = www.VNMATH.com 47. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 12 11) 1 3 1 3 4 14.2 8x x x x m+ + + + + = 12) 2 211 9 8.3 4x xx x m+ + + = HT 22: Tm m cc phng trnh sau c nghim duy nht: 1) .2 2 5 0x x m + = 2) .16 2.81 5.36x x x m + = 3) ( ) ( )5 1 5 1 2 x x x m+ + = 4) 7 3 5 7 3 5 8 2 2 x x m + + = 5) 3 4 2 3x x m+ + = 6) 9 3 1 0x x m+ + = HT 23: Tm m cc phng trnh sau c 2 nghim tri du: 1) 1 ( 1).4 (3 2).2 3 1 0x x m m m+ + + + = 2) 2 49 ( 1).7 2 0x x m m m+ + = 3) 9 3( 1).3 5 2 0x x m m+ + = 4) ( 3).16 (2 1).4 1 0x x m m m+ + + + = 5) ( )4 2 1 2 +3 8. 0x x m m + = 6) 4 2 6x x m + = HT 24: Tm m cc phng trnh sau: 1) .16 2.81 5.36x x x m + = c 2 nghim dng phn bit. 2) 16 .8 (2 1).4 .2x x x x m m m + = c 3 nghim phn bit. 3) 2 2 2 4 2 6x x m+ + = c 3 nghim phn bit. 4) 2 2 9 4.3 8x x m + = c 3 nghim phn bit. 48. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 13 VN V: PHNG TRNH LOGARIT 1. Phng trnh logarit c bn Vi a > 0, a 1: log b a x b x a= = 2. Mt s phng php gii phng trnh logarit 1) a v cng c s Vi a > 0, a 1: ( ) ( ) log ( ) log ( ) ( ) 0 ( ( ) 0)a a f x g x f x g x f x hoac g x == > > 2) M ho Vi a > 0, a 1: log ( ) log ( ) a f x b a f x b a a= = 3) t n ph 4) S dng tnh n iu ca hm s 5) a v phng trnh c bit 6) Phng php i lp Ch : Khi gii phng trnh logarit cn ch iu kin biu thc c ngha. Vi a, b, c > 0 v a, b, c 1: log logb b c a a c= Bi tp c bn HT 25: Gii cc phng trnh sau (a v cng c s hoc m ho): 1) 2 log ( 1) 1x x = 2) 2 2 log log ( 1) 1x x+ = 3) 2 1/8 log ( 2) 6.log 3 5 2x x = 4) 2 2 log ( 3) log ( 1) 3x x + = 5) 4 4 4 log ( 3) log ( 1) 2 log 8x x+ = 6) lg( 2) lg( 3) 1 lg5x x + = 7) 8 8 2 2log ( 2) log ( 3) 3 x x = 8) lg 5 4 lg 1 2 lg0,18x x + + = + 9) 2 3 3 log ( 6) log ( 2) 1x x = + 10) 2 2 5 log ( 3) log ( 1) 1/ log 2x x+ + = 11) 4 4 log log (10 ) 2x x+ = 12) 5 1/5 log ( 1) log ( 2) 0x x + = www.VNMATH.com 49. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 14 13) 2 2 2 log ( 1) log ( 3) log 10 1x x + + = 14) 9 3 log ( 8) log ( 26) 2 0x x+ + + = HT 26: Gii cc phng trnh sau (a v cng c s hoc m ho): 1) 3 1/33 log log log 6x x x+ + = 2) 2 2 1 lg( 2 1) lg( 1) 2lg(1 )x x x x+ + + = 3) 4 1/16 8 log log log 5x x x+ + = 4) 2 2 2 lg(4 4 1) lg( 19) 2lg(1 2 )x x x x+ + + = 5) 2 4 8 log log log 11x x x+ + = 6) 1/2 1/2 1/ 2 log ( 1) log ( 1) 1 log (7 )x x x + + = + 7) 2 2 3 3 log log log logx x= 8) 2 3 3 2 log log log logx x= 9) 2 3 3 2 3 3 log log log log log logx x x+ = 10) 2 3 4 4 3 2 log log log log log logx x= HT 27: Gii cc phng trnh sau (a v cng c s hoc m ho): 1) 2 log (9 2 ) 3x x = 2) 3 log (3 8) 2x x = 3) 7 log (6 7 ) 1x x + = + 4) 1 3 log (4.3 1) 2 1x x = 5) 5 log (3 ) 2 log (9 2 ) 5 xx = 6) 2 log (3.2 1) 2 1 0x x = 7) 2 log (12 2 ) 5x x = 8) 5 log (26 3 ) 2x = 9) 1 2 log (5 25 ) 2x x+ = 10) 1 4 log (3.2 5)x x+ = 11) 1 1 6 log (5 25 ) 2x x+ = 12) 1 1 5 log (6 36 ) 2x x+ = HT 28: Gii cc phng trnh sau (a v cng c s hoc m ho): 1) 2 5 log ( 2 65) 2x x x + = 2) 2 1 log ( 4 5) 1x x x + = 3) 2 log (5 8 3) 2x x x + = 4) 3 2 1 log (2 2 3 1) 3x x x x+ + + = 5) 3 log ( 1) 2x x = 6) log ( 2) 2x x + = 7) 2 2 log ( 5 6) 2x x x + = 8) 2 3 log ( ) 1x x x+ = 9) 2 log (2 7 12) 2x x x + = 10) 2 log (2 3 4) 2x x x = 11) 2 2 log ( 5 6) 2x x x + = 12) 2 log ( 2) 1x x = 13) 2 3 5 log (9 8 2) 2x x x+ + + = 14) 2 2 4 log ( 1) 1x x+ + = 15) 15 log 2 1 2x x = 16) 2log (3 2 ) 1 x x = 50. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 15 17) 2 3 log ( 3) 1 x x x + + = 18) 2 log (2 5 4) 2x x x + = HT 29: Gii cc phng trnh sau (t n ph): 1) 2 2 3 3 log log 1 5 0x x+ + = 2) 2 2 1/22 log 3log log 2x x x+ + = 3) 4 7 log 2 log 0 6x x + = 4) 2 2 1 2 2 log 4 log 8 8 x x + = 5) 2 2 1/22 log 3log log 0x x x+ + = 6) 2 2 log 16 log 64 3xx + = 7) 5 1 log log 2 5x x = 8) 7 1 log log 2 7x x = 9) 5 1 2log 2 log 5x x = 10) 2 2 3 log log 4 0x x = 11) 3 3 3 log log 3 1 0x x = 12) 3 3 2 2 log log 4 / 3x x+ = 13) 3 3 2 2 log log 2 / 3x x = 14) 2 2 4 1 log 2log 0x x + = 15) 2 2 1/4 log (2 ) 8 log (2 ) 5x x = 16) 2 5 25 log 4 log 5 5 0x x+ = 17) 29 log 5 log 5 log 5 4x x x x+ = + 18) 2 9 log 3 log 1 x x+ = 19) 1 2 1 4 lg 2 lgx x + = + 20) 1 3 1 5 lg 3 lgx x + = + 21) 2 3 2 16 4 log 14log 40log 0x x x x x x + = HT 30: Gii cc phng trnh sau (t n ph): 1) 2 33 log ( 12)log 11 0x x x x+ + = 2) 22 2log log 6 6.9 6. 13. x x x+ = 3) 2 2 2 .log 2( 1).log 4 0x x x x + + = 4) 2 2 2 log ( 1)log 6 2x x x x+ = 5) 2 3 3 ( 2)log ( 1) 4( 1)log ( 1) 16 0x x x x+ + + + + = 6) 2 2 log (2 ) log 2 xx x x + + = 7) 2 3 3 log ( 1) ( 5)log ( 1) 2 6 0x x x x+ + + + = 8) 3 3 4 log 1 log 4x x = 9) 2 2 2 2 2 log ( 3 2) log ( 7 12) 3 log 3x x x x+ + + + + = + HT 31: Gii cc phng trnh sau (t n ph): 1) 7 3 log log ( 2)x x= + 2) 2 3 log ( 3) log ( 2) 2x x + = www.VNMATH.com 51. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 16 3) 3 5 log ( 1) log (2 1) 2x x+ + + = 4) ( )6 log 2 6 log 3 log x x x+ = 5) ( )7 log 3 4 x x + = 6) ( )2 3 log 1 logx x+ = 7) 2 2 2 log 9 log log 32 .3 x x x x= 8) 2 2 3 7 2 3 log (9 12 4 ) log (6 23 21) 4x x x x x x+ + + + + + + = 9) ( ) ( ) ( )2 2 2 2 3 6 log 1 .log 1 log 1x x x x x x + = HT 32: Gii cc phng trnh sau (s dng tnh n iu): 1) 2 2 log 3 log 5 ( 0)x x x x+ = > 2) 2 2 log log2 3 5 x x x + = 3) 5 log ( 3) 3x x+ = 4) 2 log (3 )x x = 5) 2 2 2 log ( 6) log ( 2) 4x x x x + = + + 6) 2 log 2.3 3 x x + = 7) 2 3 4( 2) log ( 3) log ( 2) 15( 1)x x x x + = + HT 33: Gii cc phng trnh sau (a v phng trnh tch): 1) 2 7 2 7 log 2.log 2 log .logx x x x+ = + 2) 2 3 3 2 log .log 3 3.log logx x x x+ = + 3) ( ) ( )x 2 9 3 3 2 log log .log 2 1 1x x= + HT 34: Gii cc phng trnh sau (phng php i lp): 1) 2 3 ln(sin ) 1 sin 0x x + = 2) ( )2 2 2 log 1 1x x x+ = 3) 2 1 3 2 2 3 8 2 2 log (4 4 4) x x x x + + = + HT 35: Tm m cc phng trnh sau: 1) ( )2 log 4 1x m x = + c 2 nghim phn bit. 2) 2 3 3 log ( 2).log 3 1 0x m x m + + = c 2 nghim x1, x2 tho x1.x2 = 27. 3) 2 2 2 2 4 2 2log (2 2 4 ) log ( 2 )x x m m x mx m + = + c 2 nghim x1, x2 tho 2 2 1 2 1x x+ > . 4) 2 2 3 3 log log 1 2 1 0x x m+ + = c t nht mt nghim thuc on 3 1;3 . 5) ( ) 2 2 2 4 log log 0x x m+ + = c nghim thuc khong (0; 1). 52. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 17 VN VI: H PHNG TRNH M V LOGARIT Khi gii h phng trnh m v logarit, ta cng dng cc phng php gii h phng trnh hc nh: Phng php th. Phng php cng i s. Phng php t n ph. . HT 36: Gii cc h phng trnh sau: 1) 2 5 2 1 y y x x + = = 2) 2 4 4 32 x x y y = = 3) 2 3 1 3 19 y y x x = + = 4) 1 2 6 8 4 y y x x = = HT 37: Gii cc h phng trnh sau: 1) 4 3 7 4 .3 144 x y x y = = 2) 2 3 17 3.2 2.3 6 x y x y + = = 3) 1 2 2.3 56 3.2 3 87 x yx x yx + + + + = + = 4) 2 2 2 2 1 3 2 17 2.3 3.2 8 x y x y + + + + = + = 5) 1 1 1 3 2 4 3 2 1 x y x y + + + = = 6) 2 2 2 2( 1) 1 2 2 1. 4 4.4 .2 2 1 2 3.4 .2 4 x x y y y x y + = = 7) 2 cot 3 cos 2 y y x x = = 8) 2 2 2 2 ( )2 1 9( ) 6 y x x y x y x y + = + = 9) 2 3 2 77 3 2 7 x y x y = = 10) 2 2 2 2 ( )( 2) 2 x y y x xy x y = + + = HT 38: Gii cc h phng trnh sau: 1) 3 2 1 3 2 1 x y y x = + = + 2) 3 2 11 3 2 11 x y x y y x + = + + = + 3) 2 2 2 2 3 x y y x x xy y = + + = 4) 1 1 7 6 5 7 6 5 x y y x = = HT 39: Gii cc h phng trnh sau: 53. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 18 1) 2 2 6 log log 3 x y x y + = + = 2) log log 2 6 yx y x x y + = + = 3) 2 2 log 4 2 log 2 x y x y + = = 4) ( ) ( ) 2 2 3 5 3 log log 1 x y x y x y = + = 5) 32 log 4y xy x = = 6) 2 3 log log 2 3 9 y y x x + = = 7) 2(log log ) 5 8 y x x y xy + = = 8) 2 3 9 3 1 2 1 3log (9 ) log 3 x y x y + = = 9) 2 3 3 3 2 1 log log 0 2 2 0 x y x y y = + = 10) 3 12 log 1 3y y x x = = HT 40: Gii cc h phng trnh sau: 1) ( ) ( ) log 3 2 2 log 2 3 2 x y x y x y + = + = 2) log (6 4 ) 2 log (6 4 ) 2 x y x y y x + = + = 3) 2 2 3 3 2 2 log 1 2 log log log 4 x y y x y = + = 4) 2 2 4 4 log log 1 log log 1 y x y x y = = 5) ( )2 2 2 3 3 log 6 4 log log 1 x y x y + + = + = 6) 2 2 2 2 log log 16 log log 2 y x x y x y + = = 7) 3 3log log 3 3 2. 27 log log 1 y x x y y x + = = 8) 2 2 2 4 2 log log 3. 2. 10 log log 2 y x x y x y + = + = 9) ( ) ( ) log 2 2 2 log 2 2 2 x y x y y x + = + = 10) ( )2 2 log 4 log 2 xy x y = = HT 41: Gii cc h phng trnh sau: 1) lg lg lg 4 1000y x y x + = = 2) ( ) 2 6 36 4 2 log 9 x y x x y x = + = 54. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 19 3) 5 5 ( )3 27 3log ( ) y x x y x y x y + = + = 4) lg lg lg 4 lg 3 3 4 (4 ) (3 ) x y x y = = 5) 21 2 2 log 2log 5 0 32 x y x y xy + = = HT 42: Gii cc h phng trnh sau: 1) 2 log 4 2 2 2 log log 1 x y x y = = 2) ( ) ( ) ( ) 2 2 2 1 3 3 log log 4 x y x y x y x y = + + = 3) 8 8log log 4 4 4 log log 1 y x x y x y + = = 4) ( )1 3 3 .2 18 log 1 x y x y = + = 5) ( ) 2 2 2 1 3 3 log ( ) log ( ) 4 x y x y x y x y = + + = 6) ( ) ( )3 3 4 32 log 1 log x y y x x y x y + = = + 7) ( )3 3 .2 972 log 2 x y x y = = 8) ( )5 3 .2 1152 log 2 x y x y = + = 9) ( ) ( ) 2 2 log log 1 x y x y x y x y + = = 10) 3 3 log log 2 2 2 4 2 ( ) 3 3 12 xy xy x y x y = + + = 11) 3 3log log 3 3 2 27 log log 1 y x x y y x + = = 12) 2 2 log log log 4 3y x y x xy x y y = = + 55. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 20 VN VII: BT PHNG TRNH M Khi gii cc bt phng trnh m ta cn ch tnh n iu ca hm s m. ( ) ( ) 1 ( ) ( ) 0 1 ( ) ( ) f x g x a f x g x a a a f x g x > >> < < < Ta cng thng s dng cc phng php gii tng t nh i vi phng trnh m: a v cng c s. t n ph. . Ch : Trong trng hp c s a c cha n s th: ( 1)( ) 0M N a a a M N> > HT 43: Gii cc bt phng trnh sau (a v cng c s): 1) 2 1 2 1 3 3 x x x x 2) 6 3 2 1 1 1 1 2 2 x x x + < 3) 2 3 4 1 2 2 2 2 5 5 x x x x x+ + + + + > 4) 1 2 3 3 3 11 x xx + < 5) 2 2 3 2 3 2 9 6 0x x x x + + < 6) 2 3 7 3 1 6 2 .3x x x+ + < 7) 2 2 212 2 4 .2 3.2 .2 8 12 x x x x x x x + + + > + + 8) 2 1 2 6. 3 . 3 2.3 . 3 9x x x x x x x+ + + < + + 9) 1 2 1 2 9 9 9 4 4 4x x x x x x+ + + + + + < + + 10) 1 3 4 2 7.3 5 3 5x x x x+ + + + + + 11) 2 1 2 2 5 2 5x x x x+ + + + < + 12) 1 2 2 3 36.x x + > 13) ( ) ( ) 3 1 1 3 10 3 10 3 x x x x + + + < 14) ( ) ( ) 1 1 2 1 2 1 x x x + + 15) 2 1 2 1 2 2 x x x 16) 1 1 2 1 3 1 2 2x x + HT 44: Gii cc bt phng trnh sau (t n ph): 1) 2.14 3.49 4 0x x x + 2) 1 1 1 2 4 2 3 0x x 3) 2 ( 2)2( 1) 34 2 8 52 xxx + > 4) 4 4 1 8.3 9 9x x x x+ + + > 56. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 21 5) 25.2 10 5 25x x x + > 6) 2 1 1 5 6 30 5 .30 x x x x+ + + > + 7) 6 2.3 3.2 6 0x x x + 8) 27 12 2.8x x x + > 9) 1 1 1 49 35 25x x x 10) 1 2 1 23 2 12 0 x x x+ + < 11) 2 2 2 2 1 2 1 2 25 9 34.25x x x x x x + + + 12) 2 4 4 3 8.3 9.9 0x x x x+ + + > 13) 1 1 1 4 5.2 16 0x x x x+ + + + 14) ( ) ( )3 2 3 2 2 x x + + 15) 2 1 1 1 1 3 12 3 3 x x + + > 16) 3 1 1 1 128 0 4 8 x x 17) 1 1 1 2 2 2 9x x + + < 18) ( ) 22 1 2 9.2 4 . 2 3 0x x x x+ + + HT 45: Gii cc bt phng trnh sau (s dng tnh n iu): 1) 22 3 1 x x < + 2) 1 2 2 1 0 2 1 x x x + 3) 2 2.3 2 1 3 2 x x x x + 4) 4 2 4 3 2 13x x+ + + > 5) 2 3 3 2 0 4 2 x x x + 6) 2 3 4 0 6 x x x x + > 7) ( ) 22 2 x 3x 2x 3 .2x 3x 2x 35 2 5 2x x x + + > + + HT 46: Tm m cc bt phng trnh sau c nghim: 1) 4 .2 3 0x x m m + + 2) 9 .3 3 0x x m m + + 3) 2 7 2 2x x m+ + 4) ( ) ( ) 2 2 1 2 1 2 1 0 x x m + + + = HT 47: Tm m cc bt phng trnh sau nghim ng vi: 1) (3 1).12 (2 ).6 3 0x x x m m+ + + < , x > 0. 2) 1 ( 1)4 2 1 0x x m m+ + + + > , x. 3) ( ).9 2 1 6 .4 0x x x m m m + + , x [0; 1]. 4) 2 .9 ( 1).3 1 0x x m m m+ + + > , x. 5) ( )cos cos 2 4 2 2 1 2 4 3 0 x x m m+ + + < , x. 6) 1 4 3.2 0x x m+ , x. 7) 4 2 0x x m , x (0; 1) 8) 3 3 5 3x x m+ + , x. 9) 2.25 (2 1).10 ( 2).4 0x x x m m + + + , x 0. 10) 1 4 .(2 1) 0x x m + > , x. HT 48: Tm m mi nghim ca (1) u l nghim ca bt phng trnh (2): 57. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 22 1) ( ) ( ) 2 1 1 2 2 1 1 3 12 (1) 3 3 2 3 6 1 0 (2) x x m x m x m + + > < 2) 2 1 1 2 2 2 2 8 (1) 4 2 ( 1) 0 (2) x x x mx m + > < VN VIII: BT PHNG TRNH LOGARIT Khi gii cc bt phng trnh logarit ta cn ch tnh n iu ca hm s logarit. 1 ( ) ( ) 0 log ( ) log ( ) 0 1 0 ( ) ( ) a a a f x g x f x g x a f x g x > > >> < < < < Ta cng thng s dng cc phng php gii tng t nh i vi phng trnh logarit: a v cng c s. t n ph. . Ch : Trong trng hp c s a c cha n s th: log 0 ( 1)( 1) 0a B a B> > ; log 0 ( 1)( 1) 0 log a a A A B B > > HT 49: Gii cc bt phng trnh sau (a v cng c s): 1) 5 5 log (1 2 ) 1 log ( 1)x x < + + 2) ( )2 9 log 1 2log 1x < 3) ( ) 1 1 3 3 log 5 log 3x x < 4) 2 1 5 3 log log log 0x > 5) 1 2 3 1 2 log (log ) 0 1 x x + > + 6) ( )2 1 2 4 log 0x x > 7) ( )2 1 4 3 log log 5 0x > 8) 2 6 6log log 6 12 x x x+ 9) ( ) ( )2 2 log 3 1 log 1x x+ + 10) ( ) 2 2 2 log log 2 x x x+ 11) 3 1 2 log log 0x 12) 8 1 8 2 2log ( 2) log ( 3) 3 x x + > 58. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 23 13) ( ) ( )2 2 1 5 3 1 3 5 log log 1 log log 1x x x x + + > + HT 50: Gii cc bt phng trnh sau: 1) ( ) ( ) 2 lg 1 1 lg 1 x x < 2) ( ) ( ) 2 3 2 3 2 log 1 log 1 0 3 4 x x x x + + > 3) ( )2 lg 3 2 2 lg lg2 x x x + > + 4) 2 2 log 5 log 2 log 18 0x x x x x + < 5) 2 3 1 log 0 1 x x x > + 6) 2 3 2 3 2 log .log log log 4 x x x x< + 7) 4 log (log (2 4)) 1x x 8) 2 3 log (3 ) 1 x x x > 9) ( )2 5 log 8 16 0x x x + 10) ( )2 2 log 5 6 1x x x + < 11) 6 2 3 1 log log 0 2x x x+ > + 12) ( ) ( )21 1 log 1 log 1x x x x + > + 13) 2 3 (4 16 7).log ( 3) 0x x x + > 14) 2 (4 12.2 32).log (2 1) 0x x x + HT 51: Gii cc bt phng trnh sau (t n ph): 1) 2 log 2log 4 3 0x x + 2) ( ) ( )5 5 log 1 2 1 log 1x x < + + 3) 5 2log log 125 1x x < 4) 22 log 64 log 16 3x x + 5) 2 2 log 2.log 2.log 4 1x x x > 6) 2 2 1 1 2 4 log log 0x x+ < 7) 4 2 2 2 2 2 log log2 1 log 1 log 1 log x x x x x + > + 8) 2 2 1 2 1 4 log 2 logx x + + 9) 2 1 2 2 log 6log 8 0x x + 10) 2 3 3 3 log 4log 9 2log 3x x x + 11) 2 2 9 3 log (3 4 2) 1 log (3 4 2)x x x x+ + + > + + 12) 5 5 1 2 1 5 log 1 logx x + < + 13) 2 1 1 8 8 1 9log 1 4 logx x > 14) 100 1 log 100 log 0 2x x > 15) 2 3 3 1 log 1 1 log x x + > + 16) 216 1 log 2.log 2 log 6x x x > HT 52: Gii cc bt phng trnh sau (s dng tnh n iu): 1) 2 log0,5 0,5 ( 1) (2 5)log 6 0x x x x+ + + + 2) 2 3 log (2 1) log (4 2) 2x x + + + 3) ( ) ( )2 3 3 2 log 1 log 1x x > + + 4) 5 lg 5 0 2 3 1x x x x + < + HT 53: Tm m cc bt phng trnh sau c nghim: 1) ( )2 1/2 log 2 3x x m + > 2) 1 log 100 log 100 0 2x m > 59. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 24 3) 1 2 1 5 log 1 logm m x x + < + 4) 2 1 log 1 1 log m m x x + > + 5) 2 2 log logx m x+ > 6) 2 2 log ( 1) log ( 2)x m x m x x x > + HT 54: Tm m cc bt phng trnh sau nghim ng vi: a) ( ) ( )2 2 2 2 log 7 7 log 4x mx x m+ + + , x b) ( )2 2 2 2 log 2 4 log 2 5x x m x x m + + + , x [0; 2] c) 2 2 5 5 1 log ( 1) log ( 4 )x mx x m+ + + + , x. d) 2 1 1 1 2 2 2 2 log 2 1 log 2 1 log 0 1 1 1 m m m x x m m m + + > + + + , x N TP HT 55: Gii cc phng trnh sau: 1) 2 1 1 1 2 .4 64 8 x x x + = 2) 3 1 8 2 9 3x x = 3) 0,5 0,2 (0,04) 255 x x+ = 4) 2 1 2 11 9 5 9 5 . 3 25 3 x x x+ + = 5) 2 1 11 7 .7 14.7 2.7 48 7 x x x x+ + + = 6) ( )2 7,2 3,9 3 9 3 lg(7 ) 0x x x + = 7) 2 1 1 3 22(2 ) 4 x x x + = 8) 1 5 . 8 500 xx x = 9) 21 1 lg 3 3 1 100 x x = 10) lg 2 1000x x x= 11) lg 5 5 lg3 10 x x x + + = 12) ( ) 3 log 1 3 x x = HT 56: Gii cc phng trnh sau: 1) 2 2 2 2 4 9.2 8 0x x+ + + = 2) 2 2 5 1 5 4 12.2 8 0x x x x + = 3) 64.9 84.12 27.16 0x x x + = 4) 1 3 3 64 2 12 0x x + + = 60. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 25 5) 2 2 1 3 9 36.3 3 0x x + = 6) 4 8 2 5 2 3 4.3 28 2log 2x x+ + + = 7) 2 1 2 2( 1) 3 3 1 6.3 3x x x x+ + + = + + 8) ( ) ( )5 24 5 24 10 x x + + = 9) 3 3 1 log 1 log 9 3 210 0 x x+ + = 10) 2 lg 1 lg lg 2 4 6 2.3 0x x x+ + = 11) 2 2 sin cos 2 4.2 6x x + = 12) lg(tan ) lg(cot ) 1 3 2.3 1x x + = HT 57: Gii cc bt phng trnh sau: 1) 6 5 2 52 25 5 4 x x + < 2) 1 1 2 1 2 2 1 x x + < + 3) 2 2 .5 5 0x x x + < 4) 2 lg 3 lg 1 1000x x x + > 5) 4 2 4 2 1 x x x + 6) 2 3 2 8. 1 33 2 xx x x > + 7) 2 3 4 1 2 2 2 2 5 5x x x x x+ + + + + > 8) 2 2 log ( 1) 1 1 2 x > 9) 2 21 9 3 x x + > 10) 1 2 21 1 3 27 x x + > 11) 2 1 3 11 1 5 5 x x + > 12) 72 1 1 3 . . 1 3 3 x x > HT 58: Gii cc bt phng trnh sau: 1) 2 4 2.5 10 0x x x > 2) 1 25 5 50x x + 3) 1 1 1 9.4 5.6 4.9x x x + < 4) 2 lg 2 lg 5 3 3 2x x+ + < 5) 1 4 4 16 2log 8x x+ < 6) 2 3 2 1 1 2 21. 2 0 2 x x + + + 7) 2( 2) 2( 1) 34 2 8 52 x x x + > 8) 2 3 4 3 1 3 35. 6 0 3 x x + 9) 2 9 3 3 9x x x+ > 10) 9 3 2 9 3x x x + HT 59: Gii cc phng trnh sau: 1) 3 log (3 8) 2x x = 2) 2 5 log ( 2 65) 2x x x + = 61. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 26 3) 7 7 log (2 1) log (2 7) 1x x + = 4) 3 3 log (1 log (2 7)) 1x + = 5) 3log lg 2 3 lg lg 3 0 x x x + = 6) 3 log (1 2 ) 2 9 5 5 x x = 7) 1 lg 10x x x+ = 8) ( ) 5 log 1 5 x x = 9) 2 2 lg lg 2 lg lg 2 x x x x + = 10) lg 7 lg 14 10 x x x + + = 11) 3 9 1 log log 9 2 2 x x x + + = 12) 3 3 3 3 2log 1 log 7 1 x x x x + = HT 60: Gii cc phng trnh sau: 1) ( ) 2 2 log 5 3log 5 1 0x x + = 2) 1/3 1/3 log 3 log 2 0x x + = 3) 2 2 2 log 2log 2 0x x+ = 4) 1 3 3 2log 3 2log ( 1)x x+ + = + 5) ( )2 2 3 log 9 .log 4x x x = 6) ( )2 3 1/2 1/2 log log 3log 5 2x x + = 7) 2 2 2 lg (100 ) lg (10 ) lg 6x x x + = 8) 2 2 2 2 2 9 log (2 ).log (16 ) log 2 x x x= 9) 3 3 log (9 9) log (28 2.3 )x x x+ = + 10) 1 2 2 2 log (4 4) log 2 log (2 3)x x x+ + = + HT 61: Gii cc bt phng trnh sau: 1) 2 0,5 log ( 5 6) 1x x + > 2) 7 2 6 log 0 2 1 x x > 3c) 3 3 log log 3 0x x < 4) 1/3 2 3 log 1 x x 5) 1/4 1/4 2 log (2 ) log 1 x x > + 6) 2 1/3 4 log log ( 5) 0x > 7) 2 2 1/2 4 0 log ( 1) x x < 8h) 2 log ( 1) 0 1 x x + > 9) 2 2 log ( 8 15) 2 1x x x + + < 10) 1/3 2 5 log 3 (0,5) 1 x x + + > HT 62: Gii cc h phng trnh sau: 1) 2 ( ) 1 4 1 5 125 x y x y + = = 2) 3 2 3 4 128 5 1 x y x y + = = 3) 2 2 12 5 x y x y + = + = 62. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 27 4) 3.2 2.3 2,75 2 3 0,75 x x x y + = = 5) 7 16 0 4 49 0 x x y y = = 6) 3 3 .2 972 log ( ) 2 x y x y = = 7) 5 4 3.4 16 2 12 8 x y x y y x y = = 8) 2 /2 3 2 77 3 2 7 x y x y = = 9) ( ) ( ) 2 2 2 2 2 1 9 6 y x x y x y x y + = + = HT 63: Gii cc h phng trnh sau: 1) 4 2 2 2 log log 0 5 4 0 x y x y = + = 2) 3 4 log ( ) 2 7 log log 6x x y x y = = 3) lg 2 20 y x xy = = 4) 2 2 2 4 log 2log 3 16 + = + = x y x y 5) 3 3 3 1 1 2 15 log log 1 log 5 x y x y = + = + 6) 5 7 log 2 log log 3 log 3 2 x y y x y x = = 7) 2 2 lg( ) 1 lg13 lg( ) lg( ) 3lg2 + = + = x y x y x y 8) 2 2 2 2 9 8 log log 3 x y y x x y + = + = 9) 8 2(log log ) 5y x xy x y = + = 10) 2 1 2 2 2log 3 15 3 .log 2log 3 y y y x x x + = = + 11) 3 3 4 32 log ( ) 1 log ( ) x y y x x y x y + = = + 12) 2 3 .2 576 log ( ) 4 x y y x = = HT 64: Gii cc phng trnh sau: 1) 22 1 5 5 12.2 8 0 4 x x x x + = 2) 2 3 3 ( 1)log 4 log 16 0x x x x+ = 3) 2 2 1 2 2 1 log ( 1) log ( 4) log (3 ) 2 x x x + + = 4) 2 2 3 2 log ( 2 1) log ( 2 )x x x x+ + = + 5) 2 3 2 2 2 3 2 log ( 1) logx x x x = + 6) 5 3 5 3 log .log log logx x x x= + 7) 1 2 2 log (2 1).log (2 2) 6x x+ + + = 8) 3 3 2 3 2 3 1 log .log log log 23 x x x x = + 9) 32 1 89 25 3 log log 2 2x x x x + = 10) 2 2 0,5 2 log log log 4x x x x+ = 11) 2 3 3 1 1 1 4 4 4 3 log ( 2) 3 log (4 ) log ( 6) 2 x x x+ = + + 12) 2 3 4 82 log ( 1) 2 log 4 log (4 )x x x+ + = + + 63. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 28 /s: 1) 9 ; 3 4 x x= = 2) 1 ; 3 81 x x= = 3) 11; 1 14x x= = + 4) 1 3x = 5) nh gi 1x = 6) 1; 15x x= = 7) 2 log 3 8) 3 1; 8 x x= = 9) 5 8 x = 10) 1 1 ; ; 2 4 2 x x x= = = 11) 2; 1 33x x= = 12) 2 24; 2x x= = HT 65: Gii cc bt phng trnh sau: 1) 5 2log log 125 1x x < 2) ( ) 2 2 2 log log 2 4 x x x+ 3) 2 2 2 2 1 2 4 .2 3.2 .2 8 12x x x x x x x+ + + > + + 4) 2 2 1 1 2 3 log ( 3) log ( 3) 0 1 x x x + + > + 5) 1 1 8 2 4 2 5x x x+ + + + > 6) 2 2 2 log 3 2 log 3 x x + > + 7) 4 1 4 3 1 3 log (3 1)log 16 4 x x 8) 1 1 2 2 ( 1)log (2 5).log 6 0x x x x+ + + + 9) 2 1 2 2 1 1 0 log (2 1) log 3 2x x x + > + /s: 1) ( )1 0; 1;5 5 5 x 2) (0; )x + 3) ( ) ( )2; 1 2;3x 4) ( 2; 1) 5) (0;2] 6) 1 1 ; 8 2 7)(0;1) (3; ) + 8) [(0;2] 4; ) + 9) 1 13 3 5 ;1 ; 6 2 + + + HT 66: Gii cc h phng trnh sau: 1) 2 2 log ( ) log 3 2 2 9 3 2.( ) 3 3 6 xy xy x y x y = + + = + + 2) 2 2 2 4 2 log ( ) 5 2log log 4 x y x y + = + = 3) 2 2 2 2 2 log 2 log 5 4 log 5 x x x y y y + + = + = 4) 2 2 22 2 3. 7. 6 0 3 3 lg(3 ) lg( ) 4lg2 0 x y x y x y y x + = + + = 64. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 29 5) 2 3 2 3 log 3 3 log 5 3 log 1 log 1 x y x y + = = 6) 3 3 4 32 log ( ) 1 log ( ) x y y x x y x y + = = + /s: 1) 5 17 5 17 ; 2 2 2) ( )4;4 3) (2;4);(4;2) 4)( )2;2 5)(4;81) 6) (2;1) 65. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 30 TUYN TP THI CC NM HT 67: (D 2011) ( )2 2 1 2 log (8 ) log 1 1 2 0 ( )x x x x + + + = /s: 0x = HT 68: (B 2010) 2 2 log (3 1) ( , ) 4 2 3x x y x x y y = + = /s: 1 1; 2 HT 69: (D 2010) 2 2 2 4 2 0 ( , ) 2log ( 2) log 0 x x y x y x y + + = = /s: (3;1) HT 70: (A 2009) 2 2 2 2 2 2 log ( ) 1 log ( ) ( , ) 3 81x xy y x y xy x y + + = + = /s: (2;2),( 2; 2) HT 71: (A 2008) 2 2 2 1 1 log (2 1) log (2 1) 4x x x x x + + + = /s: 2 5 4 x x = = HT 72: (B 2008) 2 0,7 6 log log 0 4 x x x + < + /s:( 4; 3) (8; ) + HT 73: (D 2008) 2 1 2 3 2 log 0 x x x + /s: 2 2;1 (2;2 2) + HT 74: (A 2007) 3 1 3 2log (4 3) log (2 3) 2x x + + /s: 3 3 4 x< HT 75: (B 2007) ( ) ( )2 1 2 1 2 2 0 x x + + = /s: 1x = HT 76: (D 2007) 2 2 1 log (4 15.2 27) 2log 0 4.2 3 x x x + + + = /s: 2 log 3x = HT 77: (A 2006) 3.8 4.12 18 2.27 0x x x x + = /s: 1x = HT 78: (B 2006) 2 5 5 5 log (4 144) 4 log 2 1 log (2 1)x x + < + + /s:2 4x< < HT 79: (D 2006) Chng minh rng vi mi 0>a h c nghim duy nht: ln(1 ) ln(1 )x y e e x y y x a = + + = HT 80: (A 2004) 1 4 4 2 2 1 log ( ) log 1 25 y x y x y = + = /s: (3;4) HT 81: (D 2003) 2 2 2 2 2 3x x x x + = /s: 1 2 x x = = HT 82: (A 2002) Cho phng trnh 2 2 3 3 log log 1 2 1 0x x m+ + = (Vi m l tham s) a. Gii phng trnh vi 2m = /s: 3 3x = 66. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 31 b. Tm m phng trnh c t nht mt nghim thuc on 3 1;3 /s: 0 2m HT 83: (B 2002) ( )3 log log (9 72) 1x x /s: 9 log 73 2x< 67. CHUYN LUYN THI I HC 2013 - 2014 PHNG TRNH LNG GIC BIN SON: LU HUY THNG H NI, 8/2013 H V TN: LP :. TRNG : 68. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 1 I B CHUYN 2: CHUYN LNG GIC KIN THC CN NH Gi tr lng gic ca gc (cung) lng gic 1. nh ngha cc gi tr lng gic Cho ( , ) OA OM . Gi s ( ; )M x y . cos sin sin tan cos 2 cos cot sin x OH y OK AT k BS k Nhn xt: , 1 cos 1; 1 sin 1 tan xc nh khi , 2 k k Z cot xc nh khi , k k Z sin( 2 ) sin k tan( ) tan k cos( 2 ) cos k cot( ) cot k 2. Du ca cc gi tr lng gic 3. Gi tr lng gic ca cc gc c bit 4. H thc c bn: 2 2 sin cos 1 ; tan cot 1. ; 2 2 2 2 1 1 1 tan ; 1 cot cos sin Phn t Gi tr lng gic I II III IV cos + + sin + + tan + + cot + + 0 00 300 450 600 900 1200 1350 1500 1800 2700 3600 sin 0 1 0 1 0 cos 1 0 1 0 1 tan 0 1 1 0 0 cot 1 0 1 0 cosin O cotang sin tang H A M K B S T 69. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 2 I B 5. Gi tr lng gic ca cc gc c lin quan c bit II. Cng thc lng gic 1. Cng thc cng 2. Cng thc nhn i sin2 2sin .cos 2 2 2 2 cos 2 cos sin 2 cos 1 1 2 sin Gc i nhau Gc b nhau Gc ph nhau Gc hn km Gc hn km H qu: 70. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 3 I B 3. Cng thc bin i tng thnh tch 4. Cng thc bin i tch thnh tng III. Phng trnh lng gic c bn (Cc trng hp c bit) 1.Phng trnh sinx = sin a) 2 sin sin ( ) 2 x k x k Z x k b) sin . ( 1 1) arcsin 2 sin ( ) arcsin 2 x a a x a k x a k Z x a k c)sin sin sin sin( ) u v u v d) sin cos sin sin 2 u v u v e) sin cos sin sin 2 u v u v Cng thc h bc Cng thc nhn ba (*) 71. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 4 I B Cc trng hp c bit: sin 0 ( ) x x k k Z sin 1 2 ( ) 2 x x k k Z sin 1 2 ( ) 2 x x k k Z 2 2 sin 1 sin 1 cos 0 cos 0 ( ) 2 x x x x x k k Z 2. Phng trnh cosx = cos a)cos cos 2 ( ) x x k k Z b) cos . ( 1 1) cos arccos 2 ( ) x a a x a x a k k Z c)cos cos cos cos( ) u v u v d) cos sin cos cos 2 u v u v e) cos sin cos cos 2 u v u v Cc trng hp c bit: cos 0 ( ) 2 x x k k Z cos 1 2 ( ) x x k k Z cos 1 2 ( ) x x k k Z 2 2 cos 1 cos 1 sin 0 sin 0 ( ) x x x x x k k Z 3. Phng trnh tanx = tan a) tan tan ( ) x x k k Z b) tan arctan ( ) x a x a k k Z c) tan tan tan tan( ) u v u v d) tan cot tan tan 2 u v u v e) tan cot tan tan 2 u v u v Cc trng hp c bit: tan 0 ( ) x x k k Z tan 1 ( ) 4 x x k k Z 4. Phng trnh cotx = cot cot cot ( ) x x k k Z cot arccot ( ) x a x a k k Z Cc trng hp c bit: cot 0 ( ) 2 x x k k Z cot 1 ( ) 4 x x k k Z 72. GV.Lu Huy Thng 0968.393.899 B HC V B - CHUYN CN S TI BN Page 5 I B 5. Mt s iu cn ch : a) Khi gii phng trnh c cha cc hm s tang, cotang, c mu s hoc cha cn bc chn, th nht thit phi t iu kin phng trnh