Chuong[1]-Bai[1] - Ung Dung Dao Ham Khao Sat - Ve Do Thi Ham So

8/6/2019 Chuong[1]-Bai[1] - Ung Dung Dao Ham Khao Sat - Ve Do Thi Ham So

1/45

Nguyn Ph Khnh Lt .

5

Chng 1NG DNG CA O HM KHO ST

V V TH CA HM S

Bi 1: TNH N IU CA HM S1.1 TM TT L THUYT

1. nh ngha :Gi s K l mt khong , mt on hoc mt na khong . Hm sf xc nhtrn Kc gi l

ng bin trn K nu vi mi ( ) ( )1 2 1 2 1 2, ,x x K x x f x f x < < ;Nghch bin trn Knu vi mi ( ) ( )1 2 1 2 1 2, ,x x K x x f x f x < > .2. iu kin cn hm sn iu :Gi shm sf c o hm trn khong I

Nu hm sf ng bin trn khong I th ( )' 0f x vi mi x I ;Nu hm sf nghch bin trn khong I th ( )' 0f x vi mi x I .3. iu kin hm sn iu :Gi s I l mt khong hoc na khong hoc mt on , f l hm s lin tctrn I v c o hm ti mi im trong ca I ( tc l im thuc I nhngkhng phi u mt ca I ) .Khi :

Nu ( )' 0f x > vi mi x I th hm sf ng bin trn khong I ;Nu ( )' 0f x < vi mi x I th hm sf nghch bin trn khong I ;Nu ( )' 0f x = vi mi x I th hm sf khng i trn khong I .Ch :

Nu hm sf lin tc trn ;a b v c o hm ( )' 0f x > trn khong( );a b th hm sf ng bin trn ;a b .

Nu hm sf lin tc trn ;a b v c o hm ( )' 0f x < trn khong( );a b th hm sf nghch bin trn ;a b .

Gi s hm sf lin tc trn on ;a b .*Nu hm sf ng bin trn khong ( );a b th n ng bin trn on

;a b .


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Nguyn Ph Khnh Lt .

6

*Nu hm sf nghch bin trn khong ( );a b th n nghch bin trn on

;a b .

* Nu hm sf khng i trn khong ( );a b th khng i trn on ;a b .4. nh l mrng

Gi s hm sf c o hm trn khong I .Nu '( ) 0f x vi x I v '( ) 0f x = ch ti mt s hu hn im thucI th hm sf ng bin trn khong I ;

Nu '( ) 0f x vi x I v '( ) 0f x = ch ti mt s hu hn im thucI th hm sf nghch bin trn khong I .

1.2 DNG TON THNG GPDng 1 : Xt chiu bin thin ca hm s .

Xt chiu bin thin ca hm s ( )y f x= ta thc hin cc bc sau:

Tm tp xc

nh D c

a hm s

.

Tnh o hm ( )' 'y f x= . Tm cc gi tr ca x thuc D ( )' 0f x = hoc ( )'f x khng xc nh( ta gi l im ti hn hm s ).

Xt du ( )' 'y f x= trn tng khong x thuc D .Da vo bng xt du v iu kin suy ra khong n iu ca hm s.V d 1: Xt chiu bin thin ca cc hm s sau:

21.

1

xy

x

+=

2 2 12.

2

x xy

x

+ =

+

Gii:

21.

1

xy

x

+=

*Hm s cho xc nh trn khong ( ) ( );1 1; + .

*Ta c:

( )2

3' 0, 1

1y x

x

-= <

*Bng bin thin:x

1 +

'y y 1

+

1


3/45

Nguyn Ph Khnh Lt .

7

Vy hm sng bin trn mi khong ( );1 v ( )1; + .2 2 1

2.2

x xy

x

+ =

+

*Hm s cho xc nh trn khong ( ) ( ); 2 2; + .

*Ta c:

( )

2

24 5' , 22

x xy xx

+=

+

5' 0

1

xy

x

= =

=

*Bng bin thin :x 5 2 1 + 'y 0 + + 0

y + +

Vy, hm sng bin trn cc khong ( )5; 2 v ( )2;1 , nghch bin trn cc

khong ( ); 5 v ( )1; + .Nhn xt:

* i vi hm s ( . 0)ax b

y a ccx d

+=

+lun ng bin hoc lun nghch

bin trn tng khong xc nh ca n.

*

i v

i hm s

2

' '

ax bx c

y a x b

+ +=

+ lun c t nht hai kho

ng

niu.

* C hai dng hm s trn khng th lun n iu trn .

Bi tp tng t:

Xt chiu bin thin ca cc hm s sau:

2 11.

1

xy

x

=

+

2 4 32.

2

x xy

x

+ +=

+

1

3. 3

x

y x

+

=

2

34.

1

xy

x=

+

2

2

4 35.

2 2 4

x xy

x x

+=

2

2

2 26. 2 1

x xy x x

+ +

=+ +

V d 2: Xt chiu bin thin ca cc hm s sau:3 21. 3 24 26y x x x = + + 4 22. 6 8 1y x x x = + +


4/45

Nguyn Ph Khnh Lt .

8

Gii:3 21. 3 24 26y x x x = + +

*Hm s cho xc nh trn .

*Ta c : 2' 3 6 24y x x= +

24

' 0 3 6 24 0

2

xy x x

x

= = + =

=

*Bng xt du ca 'y :

x 4 2 + 'y 0 + 0

+ Trn khong ( )4;2 : ' 0y y> ng bin trn khong ( )4;2 ,

+ Trn mi khong ( ) ( ); 4 , 2; + : ' 0y y< nghch bin trn cc

khong ( ); 4 , ( )2; + .Hoc ta c th trnh by :

*Hm s cho xc nh trn .*Ta c : 2' 3 6 24y x x= +

24

' 0 3 6 24 02

xy x x

x

= = + =

=

*Bng bin thin :x 4 2 + 'y 0 + 0

y +

Vy, hm sng bin trn khong ( )4;2 , nghch bin trn cc khong

( ); 4 v ( )2; + .4 22. 6 8 1y x x x = + +


*Ta c: 3 2' 4 12 8 4( 1) ( 2)y x x x x = + = +

22

' 0 4( 1) ( 2) 01

xy x x

x

= = + =

=

*Bng xt du:x 2 1 + 'y 0 + 0 +


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Nguyn Ph Khnh Lt .

9

Vy,hm sng bin trn khong ( 2; ) + v nghch bin trn khong

( ; 2) .

Nhn xt:* Ta thy ti 1x = th 0y = , nhng qua 'y khng i du.

* i vi hm bc bn 4 3 2y ax bx cx dx e = + + + + lun c t nht mt

khong ng bin v mt khong nghch bin. Do vy vi hm bc bnkhng thn iu trn .

Bi tp tng t:

Xt chiu bin thin ca cc hm s sau:3 21. 3 2y x x= +

3 22. 3 3 2y x x x = + + +

4 213. 2 14

y x x= +

4 24. 2 3y x x= +

5 345. 85

y x x= + +

5 4 21 3 36. 2 25 4 2

y x x x x = +

7 6 577. 9 7 12

5

y x x x = + +

V d 3 : Xt chiu bin thin ca cc hm s sau:21. 2y x x=

2 32. 3y x x=

23. 1y x x=

24. 1 2 3 3y x x x = + + +

Gii:21. 2y x x= .

*Hm s cho xc nh trn mi na khong ( ); 0 2; + .

*Ta c: ( ) ( )2

1' , ;0 2;

2

xy x

x x

= +

.

Hm s khng c o hm ti cc im 0, 2x x= = .Cch 1 :

+ Trn khong ( ); 0 : ' 0y < hm s nghch bin trn khong ( ); 0 ,

+ Trn khong ( )2; + : ' 0y > hm sng bin trn khong ( )2; + .Cch 2 :

Bng bin thin :

x 0 2 + 'y || || +

y


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Nguyn Ph Khnh Lt .

10

Vy , hm s nghch bin trn khong ( );0 v ng bin trn khong ( )2; +

2 32. 3y x x=

*Hm s cho xc nh trn na khong ( ;3] .

*Ta c: ( ) ( )2

2 3

3(2 )' , ;0 0;3

2 3

x xy x

x x

=

.

Hm s khng c o hm ti cc im 0, 3x x= = .

Suy ra, trn mi khong ( ); 0 v ( )0;3 : ' 0 2y x= = Bng bin thin:

x 0 2 3 + 'y || + 0 ||

y

Hm sng bin trn khong (0;2) , nghch bin trn cc khong ( ; 0) v

(2;3) .

23. 1y x x=

*Hm s cho xc nh trn on 1;1 .

*Ta c: ( )2

2

1 2' , 1;1

1

xy x

x

=

Hm s khng c o hm ti cc im 1, 1x x= = .

Trn khong ( )1;1 : 2' 0 2y x= =

Bng bin thin:x

1 2

2

2

2 1 +

'y || 0 + 0 ||

y

Hm sng bin trn khong2 2

;2 2

, nghch bin trn mi khong

21;

2

v2

;12

.


7/45

Nguyn Ph Khnh Lt .

11

24. 1 2 3 3y x x x = + + +


*Ta c:2

2 3' 1

3 3

xy

x x

+=

+ +

( )

2

22

3

2' 0 3 3 2 3 1

3 3 2 3

xy x x x x

x x x

= + + = + =

+ + = +

Bng bin thin :x 1 + 'y + 0

y

Hm sng bin trn khong ( ; 1) , nghch bin trn khong ( 1; ) + .

Bi tp tng t:


21. 2y x x=

22. 1 4 3y x x x = + +

33. 3 5y x=

3 24. 2y x x=

( ) 25. 4 3 6 1y x x= +

22 36.

3 2

x xy

x

+=

+

2

27.

3

xy

x x

+=

+

V d 4 :Xt chiu bin thin ca cc hm s sau: 2| 2 3 |y x x= Gii:

2

2

2

2 3 khi 1 3| 2 3 |

2 3 khi 1 3

x x x x y x x

x x x

= =

+ + < .


8/45

Nguyn Ph Khnh Lt .

12

Bng bin thin:x 1 1 3 + 'y || + 0 || +

y

Hm sng bin trn mi khong( 1;1) v (3; )+ , nghch bin trn mi

khong ( ; 1) v (1;3) .

Bi tp tng t:

Xt chiu bin thin ca cc hm s sau:21. 5 4y x x= +

22. 3 7 6 9y x x x = + + +

23. 1 2 5 7y x x x = + +

2 24. 7 10y x x x = + +

V d 5 :Xt chiu bin thin ca hm s sau: 2sin cos2y x x= + trn on 0;

.

Gii :

*Hm s cho xc nh trn on 0;

*Ta c: ( )' 2 cos 1 2 sin , 0;y x x x = .

Trn on 0; :

0;

cos 0' 01

sin2

x

xy

x

=

= =

5

2 6 6x x x

= = = .

Bng bin thin:x

0 6

2

5

6

'y + 0 0 + 0

y

Da vo bng bin thin suy ra : hm sng bin trn cc khong 0;6

v

5;

2 6

, nghch bin trn cc khong ;6 2

v5

;6

.

Bi tp tng t:



9/45

Nguyn Ph Khnh Lt .

13

1. sin3y x= trn khong 0;3

.

2.cotx

yx

= trn khong ( )0; .

3.

( )

1 1

sin 4 2 3 cos28 4y x x= trn khong

0; 2

.

4. 3 sin 3 cos6 3

y x x

= + +

trn on 0; .

V d 6: Chng minh rng hm s = +2sin cosy x xng bin trn on

0;

3v nghch bin trn on

;3

.

Gii :

*Hm s cho xc nh trn on 0;

*Ta c: ( ) ( )= ' sin 2 cos 1 , 0;y x x x

V ( )0; sin 0x x > nn trn ( )1

0; : ' 0 cos2 3

y x x

= = = .

+ Trn khong 0;3

: ' 0y > nn hm sng bin trn on

0;

3;

+ Trn khong ;3

: ' 0y < nn hm s nghch bin trn on

;3

.

Bi tp tng t:

1. Chng minh rng hm s ( ) ( ) ( )sin sin f x x x x x = ng bin trn

on 0;2

.

2. Chng minh rng hm s cos2 2 3y x x= + nghch bin trn .

3. Chng minh rng hm s t n2

xy a= ng bin trn cc khong ( )0; v

( );2 .

4. Chng minh rng hm s3

cos3 2

xy x= + ng bin trn khong 0;18

v

nghch bin trn khong ; .18 2


10/45

Nguyn Ph Khnh Lt .

14

Dng 2 : Ty theo tham sm kho st tnh n iu ca hm s .

V d : Ty theo m kho st tnh n iu ca hm s:

( )3 2 3 21 1

1 13 2

y x m m x m x m = + + + +

Gii:* Hm s cho xc nh trn .

* Ta c ( )2 3' 1y x m m x m = + + v ( )2

2 1m m =

+ 0m = th 2' 0,y x x= v ' 0y = ch ti im 0x = . Hm sng

bin trn mi na khong ( ; 0 v )0; + . Do hm sng bin trn .

+ 1m = th ( )2

' 1 0,y x x= v ' 0y = ch ti im 1x = . Hm s

ng bin trn mi na khong ( ;1 v )1; + . Do hm sng bintrn .

+ 0, 1m m khi 2' 0x m

yx m

==

=.

Nu 0m < hoc 1m > th 2m m< Bng xt du 'y :

x m 2m + 'y + 0 0 +

Da vo bng xt du, suy ra hm sng bin trn cc khong ( );m v

( )2;m + , gim trn khong ( )2;m m .

Nu 0 1m< < th 2m m> Bng xt du 'y :

x 2m m + 'y + 0 0 +

Da vo bng xt du, suy ra hm sng bin trn cc khong ( )2;m v

( );m + , gim trn khong ( )2;m m .

Bi tp tluyn:

Ty theo m kho st tnh n iu ca hm s:1. 3 2 31 1 3

3 2y x mx m x m = + +

2. ( ) ( )3 21 11 1 2 33 2

y m x m x x m = + + +


11/45

Nguyn Ph Khnh Lt .

15

Dng 3 : Hm sn iu trn .S dng nh l viu kin cn

Nu hm s ( )f x n iu tng trn th ( )' 0, f x x .Nu hm s ( )f x n iu gim trn th ( )' 0, f x x .V d 1 : Tm m cc hm s sau lun nghch bin trnmi khong xcnh .

3 21.

mx my

x m

+ =

+ ( )22 2 3 12.

1

x m x m y

x

+ + +=

Gii :

3 21.

mx my

x m

+ =

+

*Hm s cho xc nh trn khong ( ) ( ); ;m m + *Ta c :

( )

2

2

2 3' ,

m my x m

x m

+ =

+

.

Cch 1 :* Bng xt du 'y

m 3 1 + 'y + 0 0 +

Da vo bng xt du ta thy

Nu 3 1m < < th ' 0y < hm s nghch bin trn mi khong ( ); m ,

( );m + .Cch 2 :Hm s nghch bin trn tp xc nh khi :

( ) ( ) 2' 0, ; ; 2 3 0 3 1y x m m m m m < + + < < <

( )22 2 3 1 1 22. 2

1 1

x m x m my x m

x x

+ + + = = + +

*Hm s cho xc nh trn khong ( ) ( );1 1; + .*Ta c :

( )2

2 1' 2 , 1

1

my x

x

= +


12/45

Nguyn Ph Khnh Lt .

16

+ 1

' 0, 12

m y x < , do hm s nghch bin trn mi khong ( );1 ,

( )1; + .

+ 1

2

m > khi phng trnh ' 0y = c hai nghim1 2

1x x< < hm sng

bin trn mi khong ( )1;1x v ( )21;x , trng hp ny khng tha .

Vy1

2m tha mn yu cu ca bi ton.

Bi tp tng t:

Tm m cc hm s sau lun nghch bin trnmi khong xc nh .2 7 11

1.1

x m my

x

+ =

( ) 21 2 32.

3

m x m m y

x m

+ + =

+

( ) 21 2 13.

1

m x xy

x

+ +=

+

( )2 2 2 14.

3

x m x m y

x

+ + =

V d 2 : Tm m cc hm s sau lun nghch bin trn .

( )3 21

1. 2 2 1 3 23

y x x m x m = + + + +

( )3

2 22. ( 2) ( 2) 8 13

xy m m x m x m = + + + +

Gii:

( )3 21

1. 2 2 1 3 23

y x x m x m = + + + +

* Hm s cho xc nh trn .

* Ta c : 2' 4 2 1y x x m = + + + v c ' 2 5m = +

* Bng xt du ' m 5

2

+

' 0 +

5

2m+ = th ( )=

2

' 2 0y x vi mi x v ' 0y = ch ti im = 2x

Do hm s nghch bin trn .

52

m+ < th < ' 0,y x . Do hm s nghch bin trn .


13/45

Nguyn Ph Khnh Lt .

17

5

2m+ > th =' 0y c hai nghim ( )


14/45

Nguyn Ph Khnh Lt .

18

a 2 2 + ' + 0 0 +

+Nu 2 2a < < th ' 0y > vi mi x . Hm sy ng bin trn .+Nu 2a = th ( )2' 2y x= + , ta c : ' 0 2, ' 0, 2y x y x = = > . Hmsyng bin trn mi na khong ( ; 2 v )2; + nn hm syngbin trn .+ Tng t nu 2a = . Hm syng bin trn .+Nu 2a < hoc 2a > th ' 0y = c hai nghim phn bit

1 2,x x . Gi s

1 2x x< . Khi hm s nghch bin trn khong ( )1 2;x x ,ng bin trn mi

khong ( )1;x v ( )2;x + . Do 2a < hoc 2a > khng tho mn yucu bi ton .Vy hm syng bin trn khi v ch khi 2 2a .

( ) ( )2 3 21

2. 1 1 3 53

y a x a x x = + + + +

* Hm s cho xc nh trn .

* Ta c : ( ) ( )2 2' 1 2 1 3y a x a x = + + + v c ( )2' 2 2a a = + +

Hm sy ng bin trn khi v ch khi ( )' 0, 1y x + Xt

2 1 0 1a a = =

31 ' 4 3 ' 0 1

4a y x y x a = = + =i khng tho yu cu bi

ton. 1 ' 3 0 1a y x a = = > = i tho mn yu cu bi ton.+ Xt

2 1 0 1a a * Bng xt du '

a 1 1 2 + ' 0 + 0

+ Nu 1 2a a< > th ' 0y > vi mi x . Hm syng bin trn .

+ Nu 2a = th ( )2

' 3 1y x= + , ta c : ' 0 1, ' 0, 1y x y x = = > . Hm

syng bin trn mi na khong ( ); 1 ` 1;va + nn hm syng bin trn .


15/45

Nguyn Ph Khnh Lt .

19

+ Nu 1 2, 1a a < < th ' 0y = c hai nghim phn bit1 2,x x . Gi s

1 2x x< . Khi hm s nghch bin trn khong ( )1 2;x x ,ng bin trn mi

khong ( )1;x v ( )2;x + . Do 1 2, 1a a < < khng tho mn yu cubi ton .

Do hm sy ng bin trn khi v ch khi 1 2a a< .Vy vi 1 2a th hm sy ng bin trn .

Bi tp tng t:

Tm m cc hm s sau lun ng bin trnmi khong xc nh .

( )3 2 21

1. 3 13 2

my x x m x = +

( )3

22. 2 33

xy mx m x = + + +

( ) ( )3

23. 2 1 4 1

3

xy m m x x = + +

( ) ( ) ( )3

24. 2 2 3 5 6 23

xy m m x m x = + +

Ch :Phng php:

* Hm s ( , )y f x m = tng trn ' 0 ' 0x

y x min y

.

* Hm s ( , )y f x m = gim trn ' 0 ' 0x

y x max y

.

Ch :

1) Nu2

'y ax bx c = + + th

*

0

0' 0

0

0

a b

cy x

a

= =

>

*

0

0' 0

0

0

a b

cy x

a

= =


16/45

Nguyn Ph Khnh Lt .

20

2) Hm ng bin trn th n phi xc nh trn .


17/45

Nguyn Ph Khnh Lt .

20

Dng 4 : Hm sn iu trn tp con ca .

Phng php:

* Hm s ( , )y f x m = tng x I ' 0 min ' 0x I

y x I y

.

* Hm s ( , )y f x m = gim ' 0 max ' 0x I

x I y x I y

.

V d 1 : Tm m cc hm s sau

1. 4mxyx m

+=

+lun nghch bin khong ( );1 .

2. ( )3 23 1 4y x x m x m = + + + + nghch bin trn khong ( )1;1 .Gii :

1.4mx

yx m

+=

+lun nghch bin khong ( );1 .

*Hm s cho xc nh trn khong ( );1 .*Ta c

( )

2

2

4' ,

my x m

x m

=

+

Hm s nghch bin trn khong ( );1 khi v ch khi( )

( )' 0, ;1

;1

y x

m

<

( )

2 4 0 2 2 2 22 1

1 1;1

m m mm

m mm

< < < <


18/45

Nguyn Ph Khnh Lt .

21

x 1 1

( )'g x

( )g x 2

10

Vy 10m tho yu cu bi ton .

Cch 2 :

( )'' 6 6 f x x = +

Nghim ca phng trnh ( )'' 0f x = l 1 1x = < . Do , hm s

cho nghch bin trn khong ( )1;1 khi v ch khi ( )1

lim 10x

m g x

= .

Vy 10m tho yu cu bi ton .

Bi tp tluyn:

Tm m cc hm s sau:

1.1mx

yx m

=

lun nghch bin khong ( )2; + .

2.( )

2

2 3

x my

m x m

=

+ lun nghch bin khong ( )1;2 .

3.2 2x m

yx m

=

lun nghch bin khong ( );0 .

4. ( )21

3

m x my

x m

+=

+lun nghch bin khong ( )0;1 .

V d 2 : Tm m cc hm s sau

1. 3 22 2 1y x x mx = + ng bin trn khong ( )1; + .2. 3 2 3 2y mx x x m = + + ng bin trn khong ( )3;0 .3. ( ) ( )3 21 2 1 1

3y mx m x m x m = + + + ng bin trn khong ( )2; + .

Gii :

1. 3 22 2 1y x x mx = + ng bin trn khong ( )1; + .* Hm s cho xc nh trn khong ( )1; + .*Ta c : 2' 6 4y x x m = +


19/45

Nguyn Ph Khnh Lt .

22

Hm s cho ng bin trn khong ( )1; + khi v ch khi

( )' 0, 1;y x + ( ) 26 4 , 1g x x x m x = >

Xt hm s ( ) 26 4g x x x = lin tc trn khong ( )1; + , ta c

( ) ( )' 12 4 0, 1g x x x g x = > >

ng bin trn kho

ng ( )1;

+

v ( ) ( ) ( )21 1

lim lim 6 4 2, limxx x

g x x x g x + + +

= = = +

*Bng bin thin.x 1 +

( )'g x +

( )g x +

2

Da vo bng bin thin suy ra 2 2m m

2. 3 2 3 2y mx x x m = + + ng bin trn khong ( )3;0 .*Hm s cho xc nh trn khong ( )3;0 .*Ta c : 2' 3 2 3y mx x = + Hm s cho ng bin trn khong ( )3;0 khi v ch khi ' 0,y

( )3;0x .

Hay ( ) ( )2 22 3

3 2 3 0, 3;0 , 3;03

xmx x x m x

x

+

Xt hm s ( ) 22 3

3

xg x

x

= lin tc trn khong ( )3;0 , ta c

( ) ( ) ( )2

4

6 18' 0, 3;0

9

x xg x x g x

x

+= < nghch bin trn khong ( )3;0

v ( ) ( )3 0

4lim , lim

27x xg x g x

+

= =

*Bng bin thin.x 3 0

( )'g x

( )g x 4

27


20/45

Nguyn Ph Khnh Lt .

23

Da vo bng bin thin suy ra4

27m

3. ( ) ( )3 21 2 1 13

y mx m x m x m = + + + ng bin trn khong ( )2; + .

*Hm s cho xc nh trn khong ( )2;+ .*Ta c : ( )2' 4 1 1y mx m x m = + + Hm sng bin trn khong ( )2; + khi v ch khi

( ) ( ) ( )2' 0, 2; 4 1 1 0, 2;y x mx m x m x + + + +

( ) ( ) ( )2 24 1

4 1 4 1, 2; , 2;4 1

xx x m x x m x

x x

+ + + + + +

+ +

Xt hm s ( ) ( )24 1

, 2;4 1

xg x x

x x

+= +

+ +

( ) ( )( )

( ) ( )22

2 2 1' 0, 2;4 1

x xg x x g x

x x

+ = < + + +

nghch bin trn khong

( )2; + v ( ) ( )2

9lim , lim 0

13 xxg x g x

+ +

= =

Bng bin thin.x 2 +

( )'g x

( )g x

9

13

0

Vy9

13m tho yu cu bi ton .

Bi tp tluyn:

Tm m cc hm s sau:

1.( )2 1 12

mx m x y

x m

+ + =

ng bin trn khong ( )1; + .

2. ( ) ( ) ( )3 2 2

2 7 7 2 1 2 3y x mx m m x m m = + + ng bin trn

khong ( )2; + .


21/45

Nguyn Ph Khnh Lt .

24

3.3 21 ( 1) 3( 2) 1

3y mx m x m x = + + ng bin trn khong (2; )+ .

V d 3 : Tm m cc hm s sau :

1. 2 6 2

2

mx xy

x

+ =

+nghch bin trn na khong )2; + .

2. 3 2 2( 1) (2 3 2) (2 1)y x m x m m x m m = + + + ng bin trn na

khong )1; + .Gii :

1. 2 6 2

2

mx xy

x

+ =

+nghch bin trn na khong )2; + .

*Hm s cho xc nh trn na khong )2; + *Ta c 2 2' 3 2( 1) (2 3 2)y x m x m m = + + Hm ng bin trn na khong )2; + . )' 0, 2;y x +

)2 2( ) 3 2( 1) (2 3 2) 0, 2; f x x m x m m x = + + +

V tam thc ( )f x c 2' 7 7 7 0m m m = + > nn ( )f x c hai nghim

1 2

1 ' 1 ';

3 3

m mx x

+ + + = = .

V1 2

x x< nn 1

2

( )x x

f xx x

.

Do ) 2( ) 0 2; 2 ' 5 f x x x m +

2 2

5 5 32

2' (5 ) 2 6 0

m mm

m m m

+

.

2. 3 2 2( 1) (2 3 2) (2 1)y x m x m m x m m = + + + ng bin trn na

khong )1; + .

*Hm s cho xc nh trn na khong )1; + *Ta c 2

2

4 14'

( 2)

mx mx y

x

+ +=

+

Hm nghch bin trn na khong [1; )+ 2( ) 4 14 0 f x mx mx = + + ,

) ( )1; *x + .


22/45

Nguyn Ph Khnh Lt .

25

Cch 1: Dng tam thc bc hai

Nu 0m = khi ( )* khng tha mn.Nu 0m . Khi ( )f x c 24 14m m = Bng xt du m 0

72

+

' + 0 0 +

Nu 702

m< < th ( ) 0 f x x > , nu ( )f x c hai nghim1 2,x x th

( ) 0f x 1 2

( ; )x x x nn ( )* khng tha mn.

Nu 0m < hoc 72

m > . Khi ( ) 0f x = c hai nghim

2 2

1 2

2 4 14 2 4 14;

m m m m m m x x

m m

+ =

V 0m < hoc7

2m > 1

1 22

( ) 0x x

x x f x x x

<

Do ) 22( ) 0 1; 1 3 4 14 f x x x m m m +

2

0 14

55 14 0

mm

m m


23/45

Nguyn Ph Khnh Lt .

26

* Ta c : 2' 3 6y x x m = + + c ' 9 3m = i Nu 3m th ' 0,y x , khi hm s lun ng bin trn , do

3m khng tho yu cu bi ton .

i Nu 3m < , khi ' 0y = c hai nghim phn bit ( )1 2 1 2,x x x x < v hm s

nghch bi

n trong

on 1 2;x x

v

i

di 2 1l x x=

Theo Vi-t, ta c :1 2 1 2

2,3

mx x x x + = =

Hm s nghch bin trn on c di bng 1 1l =

( ) ( )2 2

2 1 1 2 1 2

4 91 4 1 4 1

3 4x x x x x x m m = + = = = .

Bi tp tng t:

1. Tm tt c cc tham sm hm s 3 2 23 1y x m x x m = + + nghchbin trn on c di bng 1?.

2. Tm tt c cc tham sm hm s 3 2 2 3 5y x m x mx m = + + + + ngbin trn on c di bng 3 ?.

V d 5: Tm m hm s cosy x m x = + ng bin trn .Gii:

*Hm s cho xc nh trn .* Ta c ' 1 siny m x= .Cch 1: Hm ng bin trn

' 0, 1 sin 0, sin 1, (1)y x m x x m x x

* 0m = th (1) lun ng

* 0m > th1 1

(1) sin 1 0 1x x mm m

< .

* 0m < th1 1

(1) sin 1 1 0x x R m m m

< .

Vy 1 1m l nhng gi tr cn tm.

Cch 2: Hm ng bin trn ' 0y x

1 0min ' min{1 ;1 } 0 1 1

1 0

my m m m

m

= +

+

.

Bi tp tluyn:

1. Tm m hm s ( )1 cosy x m m x = + nghch bin trn .2. Tm m hm s .sin cosy x x m x = + ng bin trn .


24/45

Nguyn Ph Khnh Lt .

27


25/45

27

Dng 5 : Sdng tnh n iu ca hm s CM bt ng thc.

a bt ng thc v dng ( ) ( ), ;f x M x a b .Xt hm s ( ) ( ), ;y f x x a b= . Lp bng bin thin ca hm s trn khong ( );a b .Da vo bng bin thin v kt lun.V d 1 : Vi 0;

2x

.Chng minh rng :

1. sin t n 2x a x x + > 2 sin2. 1

x

x< <

Gii :1. sin t n 2x a x x + >

*Xt hm s ( ) sin t n 2 f x x a x x = + lin tc trn na khong 0;2

.

*Ta c : ( ) 22 21 1' cos 2 cos 2 0, 0; 2cos cos f x x x x x x

= + > + >

( )f x l hm sng bin trn 0;2

v ( ) ( )0 , f x f > 0;

2x

hay sin t n 2 , 0;2

x a x x x

+ >

(pcm).

T bi ton trn ta c bi ton sau : Chng minh rng tam gic ABC c bagc nhn th sin sin sin tan tan tan 2A B C A B C + + + + + >

2 sin2. 1xx

< <

*Vi 0x > th sin 1xx

< (xem v d 2 )

*Xt hm s ( ) sin xf xx

= lin tc trn na khong 0;2

.

*Ta c ( ) 2.cos sin' , 0; 2x x x

f x x x

=

.

*Xt hm s ( ) . cos sing x x x x = lin trc trn on 0; 2

v c


26/45

28

( ) ( )' . sin 0, 0;2

g x x x x g x

= <

lin tc v nghch bin trn on

0;2

v ta c ( ) ( )0 0, 0;2

g x g x

< =

*T suy ra ( )( )

( )2'

' 0, 0; 2

g x

f x x f x x

= <

lin tc v nghch

bin trn na khong 0;2

, ta c ( )2

, 0;2 2

f x f x

> =

.

Bi tp tng t:

Chng minh rng vi mi 0;2

x

ta lun c:

1. tan x x> 3

2. tan3

xx x> +

3. 2sin tan 3x x x+ >

34.

2 cotsin

x

xx

2 4

3. cos 1 , 0;2 24 2

x xx x

< +

3sin

4. cos , 0;2

xx x

x

>

.

Gii :

1. sin , 0;2

x x x

*Xt hm s ( ) sin f x x x = lin tc trn on 0;2

x

*Ta c: '( ) cos 1 0 , 0;2

f x x x

=

( )f x l hm nghch bin trn

on 0;2

.

Suy ra ( ) (0) 0 sin 0;2

f x f x x x

=

(pcm).


27/45

29

3

2. sin , 0;3! 2

xx x x

>

*Xt hm s 3( ) sin6

x f x x x = + lin tc trn na khong 0;

2x

.

*Ta c: 2'( ) cos 1 "( ) sin 0 0;2 2

x f x x f x x x x = + = +

(theo

cu 1)

'( ) '(0) 0 0; ( ) (0) 0 0;2 2

f x f x f x f x

= =

3

sin , 0;3! 2

xx x x

>

(pcm).

2 4

3. cos 1 , 0;

2 24 2

x xx x

< +

*Xt hm s 2 4( ) cos 12 24

x xg x x= + lin tc trn na khong 0;

2x

.

*Ta c: 3'( ) sin 0 0;6 2

xg x x x x

= +

(theo cu

2) ( ) (0) 0 0;2

g x g x

=

2 4

cos 1 , 0;2 24 2

x x

x x

< + (pcm).

3sin

4. cos , 0;2

xx x

x

>

.

Theo kt qu cu 2, ta c:3

sin , 0;6 2

xx x x

>

332 2 2 4 6sin sin1 1 1

6 6 2 12 216

x x x x x x x

x x

> > = +

3 2 4 4 2sin1 (1 )

2 24 24 9

x x x x x

x

> + +


28/45

30

V

32 2 4sin0; 1 0 1

2 9 2 24

x x x x x

x

> > +

Mt khc, theo cu 3:2 4

1 cos , 0;2 24 2

x xx x

+ >

Suy ra3sin

cos , 0;2

xx x

x

>

(pcm).

Nhn xt: Ta csin

0 sin 0 1 (0; )2

xx x x

x

< < < < nn

3sin sin

3x x

x x

. Do , ta c kt qu sau

Chng minh rng: vi 3 , ta lun c:sin

cos 0;2

xx x

x

.

Bi tp tng t:Chng minh rng :

2 2 4

1. 1 cos 1 , 02 2 24

x x xx x < < +

2

1 12. sin 1 , 0

6x x

x x> >

V d 3 : Chng minh rng2 2 2

1 1 41 , 0;

2sinx

x x

< +

Gii :

*

Xt hm s 2 2

1 1

( ) sinf x x x=

lin tc trn n

a kho

ng 0; 2

x

.

*Ta c: 3 33 3 3 3

2 cos 2 2( cos sin )'( )

sin sin

x x x x f x

x x x x

+= + = .

Theo kt qu cu 4 - v d 2 , ta c:3

sincos , 0;

2

xx x

x

>

3 3cos sin 0 , 0; '( ) 0 , 0;2 2

x x x x f x x

+ > >

24( ) 1 , 0;2 2 f x f x

=

Do vy:2 2 2

1 1 41 , 0;

2sinx

x x

< +

(pcm).


29/45

31

Bi tp tng t:

1. Chng minh rng : ( )22

4sin

xx x

< vi mi 0;2

x

.

2. Chng minh rng : ( )22sin 412x

x x x

> + vi mi 0;2

x

.

V d 4 : Vi 02

x

< . Chng minh rng3

12.sin t n 22 2 2

xx a x

+

+ >

Gii :

*Ta c:1

sin t n2.sin t n 2 sin t n 22 2 2. 2 .2 2.2

x a xx a x x a x

+

+ =

Ta chng minh:

1 3sin t n

2 2 1 32 2 sin t n2 2

xx a x

x a x x +

+ 0;2

x

.

*Xt hm s( )

1 3sin t n

2 2

x f x x a x = + lin tc trn na khong 0;

2

.

*Ta c: ( ) 3 22 2

1 3 2 cos 3 cos 1' cos

22.cos 2cos

x x f x x

x x

+= + =

2

2

(cos 1) (2cos 1)0 , 0;

22cos

x xx

x

+=

.

( )f x ng bin trn

[0; )2

1 3( ) (0) 0 sin tan

2 2 f x f x x x = + , 0;

2x

(pcm).

V d 5 : Chng minh bt ng thc sau vi mi s t nhin 1n >

1 1 2n n

n nn n

n n+ + <

Gii :

*t ( )0;1 , *n nx n Nn

= .

*Bt ng thc cn chng minh l: ( )1 1 2, 0;1n nx x x+ + < *Xt hm ( ) 1 1 , [0;1)n n f x x x x = + +

( )( ) ( )

( )1 1

1 1 1' 0, 0;1

1 1n n

n n

f x x n

x x

= < +


30/45

32

Vy ( )f x gim trn ( )0;1 nn ( ) ( ) ( )0 2, 0;1 f x f x < = .

V d 6:

1.Cho 0x y z .Chng minh rng : x z y x y z z y x y z x

+ + + +

2. Cho , , 0x y z > .Chng minh rng:4 4 4 2 2 2 2 2 2( ) ( ) ( ) ( )x y z xyz x y z xy x y yz y z zx z x + + + + + + + + + +

Gii :

1Cho 0x y z .Chng minh rng : x z y x y z z y x y z x

+ + + +

0x y z

*Xt hm s : ( ) x z y x y z f xz y x y z x

= + + + +

.

*Ta c: 2 2 21 1 1 1

'( ) ( ) ( ) ( )( ) 0, 0

y z

f x y z x z y yz x x x= =

( )f x l hm sng bin 0x ( ) ( ) 0 f x f y = pcm.

2. Cho , , 0x y z > Chng minh rng:4 4 4 2 2 2 2 2 2( ) ( ) ( ) ( )x y z xyz x y z xy x y yz y z zx z x + + + + + + + + + + .

Khng mt tnh tng qut ta gi s: 0x y z > .*Xt hm s

4 4 4 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) f x x y z xyz x y z xy x y yz y z zx z x = + + + + + + + +

*Ta c 3 2 3 3'( ) 4 3 ( ) ( ) ( ) f x x x y z xyz yz x y z y z = + + + + + + 2"( ) 12 6 ( ) 2 f x x x y z yz = + +

"( ) 0f x > (do x y z ) 2 3 2'( ) '( ) ( ) 0f x f y z y z z y z = = nn

( )f x l hm sng

bin. 4 3 2 2 2 2( ) ( ) 2 ( ) 0f x f y z z y y z z z y = + = pcm.

V d 7:

1. Cho , , 0a b c > . Chng minh rng: 32

a b c

a b b c c a + +

+ + +.

2. Cho 0 a b c< . Chng minh rng:22 2 2 ( )

3( )

a b c c a

b c c a a b a c a

+ + +

+ + + +.


31/45

33

Gii :

1.Cho , , 0a b c > . Chng minh rng: 32

a b c

a b b c c a + +

+ + +

*t , , 1b c ax y z xyz a b c

= = = = v bt ng thc cho

1 1 1 31 1 1 2x y z + +

+ + +.

*Gi s 1 1z xy nn c: 1 1 2 21 1 1 1

z

x y xy z+ =

+ + + +

2

1 1 1 2 1 2 1( )

1 1 1 1 11 1

z tf t

x y z z t z t + + + = + =

+ + + + ++ +

vi

1t z=

*Ta c:2 2 2 2 2

2 2 2(1 )'( ) 0

(1 ) (1 ) (1 )

t tf t

t t t

=

+ + +

3( ) (1) , 1

2 f t f t = pcm.

2. Cho 0 a b c< . Chng minh rng: 22 2 2 ( )3( )

a b c c a

b c c a a b a c a

+ + +

+ + + +

*t , ,1b c x xa a

= = . Khi bt ng thc cn chng minh tr

thnh22 2 2 4

1 1 1

x x x

x x x

+ ++ +

+ + + +

2 1 2 ( 1)1 (2 2 )1

x x xx x

x

+ + + + + +

+ +

*Xt hm s 2 1 2 ( 1)( ) 1 (2 2 ), 11

x x x f x x x x

x

+ += + + + +

+ +

*Ta c:2

2(2 1) 1'( ) 2 1 2

1 ( )

x f x x

x

+ = +

+ +

2

2 +1 2'( ) ( 1) 0, 1

+1( )

x f x x

x

=

+

Nh vy hm ( )f x l ng bin do 2

1( ) ( ) 3 3 f x f

= +


32/45

34

Nhng 32 2 2

1 1 1'( ) 2 3 3 3 . . 3 0f

= + = + + =

( ) ( ) (1) 0 f x f f = pcm.

Bi tp tluyn:

1.Cho hm s ( ) 2 sin t n 3 f x x a x x = +

)a Chng minh rng hm sng bin trn na khong 0;2

.

)b Chng minh rng 2 sin t n 3x a x x + > vi mi 0;2

x

.

2.

)a Chng minh rng t na x x> vi mi 0;2

x

.

)b Chng minh rng3

t n3

xa x x> + vi mi 0;

2x

.

3.

Cho hm s ( )4

t n f x x a x

= vi mi 0;4

x

)a Xt chiu bin thin ca hm s trn on 0;4

.

)b T

suy ra r

ng

4

t nx a x

v

i mi 0; 4

x

.

4.Chng minh rng cc bt ng thc sau :)a sinx x< vi mi 0x >

)b sinx x> vi mi 0x <

)c 2

cos 12

xx > vi mi 0x

)d3

sin6

xx x> vi mi 0x >

)e3

sin6

xx x< vi mi 0x <

)f sin t n 2x a x x + > vi mi 0;2

x

.


33/45

Nguyn Ph Khnh Lt .

35

Dng 6 : Dng n iu hm s gii v bin lun phng trnh vbt phng trnh .

Ch 1 :

Nu hm s ( )y f x= lun n iu nghim cch trn D ( hoc lun ng bin

hoc lun nghch bin trn D) th s nghim ca phng trnh : ( ) f x k = s

khng nhiu hn mt v ( ) ( ) f x f y = khi v ch khi x y= .Ch 2:

Nu hm s ( )y f x= lun n iu nghim cch trn D ( hoc lun ng

bin hoc lun nghch bin trn D ) v hm s ( )y g x= lun n iu nghimngoc ( hoc lun ng bin hoc lun nghch bin ) trn D, th s nghim trn

D ca phng trnh ( ) ( ) f x g x = khng nhiu hn mt.

Nu hm s ( )y f x= c o hm n cp n trn Dv phng trnh( )( ) 0kf x = c m nghim, khi phng trnh ( 1)( ) 0kf x = c nhiu nht l

1m + nghim.

V d 1 : Gii cc phng trnh

1. 2 23 (2 9 3) (4 2)( 1 1) 0x x x x x + + + + + + + =

2.3

3 2 24 5 6 7 9 4x x x x x + = +

Gii :

1. 2 23 (2 9 3) (4 2)( 1 1) 0 (1)x x x x x + + + + + + + = Phng trnh ( ) 2 2(1) 3 (2 ( 3 ) 3) (2 1)(2 (2 1) 3) (2)x x x x + + = + + + + t 3 , 2 1, , 0u x v x u v = = + >

Phng trnh 2 2(1) (2 3) (2 3) (3)u u v v + + = + +

*Xt hm s 4 2( ) 2 3 f t t t t = + + lin tc trn khong ( )0; + *Ta c ( )3

4 2

2 3'( ) 2 0, 0

3

t t f t t f t

t t

+= + > >

+

ng bin trn khong

( )0; + .

Khi phng trnh1

(3) ( ) ( ) 3 2 15

f u f v u v x x x = = = + =


34/45

Nguyn Ph Khnh Lt .

36

Vy1

5x = l nghim duy nht ca phng trnh.

2.3

3 2 24 5 6 7 9 4x x x x x + = + .

t y =3

27 9 4y x x= + . Khi phng trnh cho

3 2

2 34 5 6

7 9 4x x x y

x x y + =

+ =

( ) ( )( )

3 23 2

33 3 2 3

4 5 64 5 6

3 4 2 1 1 *

x x x y x x x y I

y y x x x y y x x

+ = + =

+ = + + + + = + + +

( )* c dng ( ) ( ) ( )1 f y f x a = +

* Xt hm ( ) 3 , f t t t t = + *V ( ) 2' 3 1 0, f t t t = + > nn hm sng bin trn tp s thc .Khi ( ) 1a y x = +

H ( )( )3 2 3 24 5 6 4 6 5 0 * *

1 1

x x x y x x xI

y x y x

+ = + =

= + = +

Gii phng trnh ( )* * ta c tp nghim :1 5 1 5

5, ,2 2

S +

=

.

V d 2 : Chng minh rng phng trnh: =22 2 11x x c nghim duy nhtGii :

Cch 1 :

Xt hm s 22 2y x x= lin tc trn na khong ) +2; .

Ta c:( )

( )

= > +

5 8' 0, 2;

2

x xy x

x ( )2lim lim 2 2

x xy x x

+ +

= = +

Bng bin thin :x 2 +

'y +

y +

0


35/45

Nguyn Ph Khnh Lt .

37

Da vo bng bin thin ta thy th ca hm s 22 2y x x= lun ct

ng thng 11y = ti duy nht mt im. Do phng trnh

=22 2 11x x c nghim duy nht .

Cch 2:

Xt hm s ( ) 22 2 11y f x x x = = lin tc trn na khong ) +2; .

Ta c ( ) ( )2 11, 3 7f f= = . V ( ) ( ) ( )2 . 3 77 0 0 f f f x = < = c t nht

mt nghim trong khong ( )2;3 .

( )( )

( ) ( )5 8

' 0, 2;2

x x f x x f x

x

= > +

lin tc v ng bin trn khong

( )2;3 .

Vy phng trnh cho c nghim duy nht thuc khong ( )2;3 .

V d 3 : Gii bt phng trnh sau : 5 1 3 4x x + + Gii :

iu kin :1

5x

*Xt hm s ( ) 5 1 3 f x x x = + + lin tc trn na khong 1 ;5

+

*Ta c : ( )5 1 1'( ) 0 ,52 5 1 2 1

f x x f x x x

= + > >

l hm s ng bin trn na khong 1 ;5

+

v (1) 4f = , khi bt phng

trnh cho ( ) (1) 1. f x f x

Vy bt phng trnh cho c nghim l 1x .

V d 4 : Gii bt phng trnh sau5

3 3 2 2 62 1

x xx

+

Gii :

iu kin:1 3

2 2x<

*Bt phng trnh cho 53 3 2 2 6 ( ) ( ) (*)2 1

x x f x g x x

+ +

*Xt hm s 5( ) 3 3 22 1

f x x x

= +

lin tc trn na khong1 3;

2 2


36/45

Nguyn Ph Khnh Lt .

38

*Ta c :3

3 5 1 3'( ) 0, ; ( )

2 23 2 ( 2 1) f x x f x

x x

= <

l hm

nghch bin trn na on1 3;

2 2

.

Hm s ( ) 2 6g x x= + l hm ng bin trn v (1) (1) 8f g= = i Nu 1 ( ) (1) 8 (1) ( ) (*)x f x f g g x > < = = < ng

i Nu 1 ( ) (1) 8 (1) ( ) (*)x f x f g g x < > = = > v nghim.

Vy nghim ca bt phng trnh cho l:3

12

x .

V d 5 : Gii bt phng trnh sau

( 2)(2 1) 3 6 4 ( 6)(2 1) 3 2x x x x x x + + + + +

Gii :

iu kin:1

2

x .

Bt phng trnh cho ( )( 2 6)( 2 1 3) 4 *x x x + + +

i Nu 2 1 3 0 5 (*)x x lun ng.i Nu 5x >

*Xt hm s ( ) ( 2 6)( 2 1 3) f x x x x = + + + lin tc trn khong( )5;+ *Ta c:

1 1 2 6'( ) ( )( 2 1 3) 0, 5

2 2 2 6 2 1

x x f x x x

x x x

+ + += + + > >

+ +

( )f x ng bin trn khong ( )5;+ v (7) 4f = , do

( )* ( ) (7) 7 f x f x .

Vy nghim ca bt phng trnh cho l:1

72

x .

V d 6 : Gii bt phng trnh sau 3 22 3 6 16 2 3 4x x x x + + + < + Gii :

iu kin:

3 22 3 6 16 0

2 4.4 0

x x x

xx

+ + +

.

Bt phng trnh cho

( )3 22 3 6 16 4 2 3 ( ) 2 3 *x x x x f x + + + < <


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Nguyn Ph Khnh Lt .

39

*Xt hm s 3 2( ) 2 3 6 16 4 f x x x x x = + + + lin tc trn on2;4 .

*Ta c: ( )23 2

3( 1) 1'( ) 0, 2;4

2 42 3 6 16

x x f x x

xx x x

+ += + >

+ + +

( )f x

ng bin trn na khong ( )2;4 v (1) 2 3f = , do

( )* ( ) (1) 1 f x f x < < .Vy nghim ca bt phng trnh cho l: 2 1x < .

V d 7 : Chng minh rng 4 1 0 ,x x x + > Gii :

*Xt hm s 4( ) 1 f x x x = + lin tc trn .*Ta c 3'( ) 4 1 f x x = v

3

1'( ) 0

4

f x x = = .

*V '( )f x i du t m sang dng khi x qua3

1

4, do

3 3 3

1 1 1min ( ) ( ) 1 0

4 4 4 4 f x f = = + >

Vy ( ) 0 , f x x > .

V d 8 : Chng minh rng phng trnh : 52

2009 0

2

xx

x

+ =

c

ng hai nghim dng phn bit. Gii :

*iu kin: 2x > (do 0x > ).*Xt hm s : 5

2

( ) 2009

2

x f x x

x

= +

vi 2x > .

*Ta c 42 3

1'( ) 5

( 2)

f x x

x

=

3

2 5

3"( ) 20 0 , 2

( 2)

x f x x x

x

= + > >

'( ) 0f x = c nhiu nht mt nghim ( ) 0f x = c nhiu nht l hainghim.


38/45

Nguyn Ph Khnh Lt .

40

M:

2

lim ( ) , ( 3) 0, lim ( ) ( ) 0x

x

f x f f x f x + +

= + < = + = c hai nghim

( )1 2; 3x v 2 3x > .V d 9 : Gii h phng trnh

1. 2 3 4 4 (1)

2 3 4 4 (2)

x y

y x

+ + =+ + =

2.( )( )

3

3

2 1

2 2

x x y

y y x

+ =

+ =

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =

Gii :

1.2 3 4 4 (1)

2 3 4 4 (2)

x y

y x

+ + =

+ + =

iu kin:

34

23

42

x

y

.

Cch 1:

Tr(1) v (2) ta c:

( )2 3 4 2 3 4 3x x y y + = + *Xt hm s

( ) 2 3 4 f t t t = + lin tc trn on

3

; 42

.

*Ta c:/ 1 1 3( ) 0, ; 4

22 3 2 4 f x t

t t

= + >

+ (3) ( ) ( )f x f y x y = = .

Thay x y= vo (1) ,ta c:

2 3 4 4 7 2 (2 3)(4 ) 16x x x x x + + = + + + =

22

39 02 2 5 12 9 11

9 38 33 0 9

xxx x x

x x x

= + + =

+ = =


39/45

Nguyn Ph Khnh Lt .

41

Vy h phng trnh c 2 nghim phn bit

113

9,3 11

9

xx

yy

==

= =

.

Cch 2:

Tr(1) v (2) ta c:

( ) ( )2 3 2 3 4 4 0x y y x + + + =

(2 3) (2 3) (4 ) (4 )0

2 3 2 3 4 4

x y y x

x y y x

+ + + =

+ + + +

( )2 1

( ) 0 *2 3 2 3 4 4

x yx y y x

+ =

+ + + + .

V2 1

02 3 2 3 4 4x y y x

+ >

+ + + +

nn ( )* x y =

Thay x y= vo (1) ,ta c:

2 3 4 4 7 2 (2 3)(4 ) 16x x x x x + + = + + + =

22

39 02 2 5 12 9 11

9 38 33 09

xxx x x

x x x

= + + = + = =

Vy h phng trnh c 2 nghim phn bit

113

9,3 11

9

xx

yy

==

= =

.

2.( )( )

3

3

2 1

2 2

x x y

y y x

+ =

+ =

Cch 1 :

*Xt hm s 3 2/( ) 2 ( ) 3 2 0, f t t t f t t t = + = + > .*H phng trnh trthnh ( ) (1)

( ) (2)

f x y

f y x

=

=.

+ Nu ( ) ( )x y f x f y y x > > > (do (1)v (2)dn n mu thun).

+ Nu ( ) ( )x y f x f y y x < < < (mu thun).

Suy ra x y= , th vo h ta c

( )3 2 20 1 0 0 1 0.x x x x x v x + = + = = + >


40/45

Nguyn Ph Khnh Lt .

42

Vy h c nghim duy nht0

0

x

y

=

=.

Cch 2:Tr(1) v (2) ta c:

3 3 2 23 3 0 ( )( 3) 0x y x y x y x y xy + = + + + = 2 23

( ) 3 02 4

y yx y x x y

+ + + = =

Thx y= vo (1) v (2) ta c: ( )3 20 1 0 0x x x x x + = + = =

Vy h phng trnh c nghim duy nht0

0

x

y

=

=.

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =

T(1) v (2) suy ra 1 , 1x y

(1) ( ) ( ) (*)f x f y =

*Xt hm s 3( ) 3 f t t t = lin tc trn on [ 1;1] , ta c( )2'( ) 3( 1) 0 [ 1;1] f t t t f t = nghch bin trn on [ 1;1]

*Do : (*) x y = thay vo (2) ta c nghim ca h l:

6

1

2x y= = .

V d 10 : Gii h phng trnh

1.2

1 1(1)

2 1 0 (2)

x yx y

x xy

=

=

2.3

1 1(1)

2 1 (2)

x yx y

y x

=

= +

Gii :

1.2

1 1(1)

2 1 0 (2)

x yx y

x xy

=

=

iu kin: 0, 0x y . Ta c:


41/45

Nguyn Ph Khnh Lt .

43

1(1) ( ) 1 0 1

.

y x

x yxy y

x

= + = =

i y x= phng trnh 2(2) 1 0 1x x = = .

i 1

y x= phng trnh (2)v nghim.

Vy h phng trnh c 2 nghim phn bit1 1;

1 1

x x

y y

= =

= = .

Bnh lun:

Cch gii sau y sai:2

1 1(1)

2 1 0 (2)

x yx y

x xy

=

=

.

iu kin: 0, 0x y .*Xt hm s

2

/1 1( ) , \ {0} ( ) 1 0, \ {0} f t t t f t t t t

= = + > .

Suy ra (1) ( ) ( ) f x f y x y = = !

Sai do hm s ( )f t n iu trn 2 khong ri nhau (c th

( ) ( )1 1 0f f = = ).

2.3

1 1(1)

2 1 (2)

x yx y

y x

=

= +

Cch 1:iu kin: 0, 0.x y

1(1) 0 ( ) 1 0 1

.

x yx y

x y x y xy xy y

x

= + = + = =

i x y= phng trnh (2)1

1 5.

2

x

x

=

=

i 1

y x= phng trnh (2) 4 2 0.x x + + =

Xt hm s 4 / 33

1( ) 2 ( ) 4 1 0 .

4 f x x x f x x x

= + + = + = =


42/45

Nguyn Ph Khnh Lt .

44

3 3

1 32 0, lim lim

4 4 4 x xf

+

= > = = +

4( ) 0, 2 0 f x x x x > + + = v nghim.

Cch 2:iu kin: 0, 0.x y

1(1) 0 ( ) 1 0 1

.

x yx y

x y x y xy xy y

x

= + = + = =

i x y= phng trnh (2)1

1 5.

2

x

x

=

=

i 1

yx

= phng trnh (2) 4 2 0.x x + + =

i Vi 41 2 0 2 0x x x x < + > + + > .

i Vi 4 41 2 0x x x x x x + + > .

Suy ra phng trnh (2)v nghim.Vy h phng trnh c 3 nghim phn bit

1 5 1 51

2 21 1 5 1 5

2 2

x xx

yy y

+ = = =

= + = =

.

Bi tp tluyn:

1. Gii phng trnh:

. 3 1 7 2 4a x x x + + + + = 3 3. 5 1 2 1 4b x x x + + =

2. Gii phng trnh: ( )10 1081

81sin cos *256

x x+ =

3. Gii bt phng trnh:

( 2)(2 1) 3 6 4 ( 6)(2 1) 3 2x x x x x x + + + + +

4. Gii cc h phng trnh

1.

2

2

2

2

1 2

12

1

xy

xyz

y

zx

z

=

=

=

2.

3 2

3 2

3 2

9 27 27 0

9 27 27 09 27 27 0

y x x

z y yx z z

+ =

+ =

+ =


43/45

Nguyn Ph Khnh Lt .

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Dng 7 : Dng n iu hm s gii v bin lun phng trnh vbt phng trnh cha tham s.

Cho hm s ( ); 0f x m = xc nh vi mi ( )*x I Bin i ( )* v dng ( ) ( ) f x f m = Xt hm s ( )y f x= lin tc trn IDng tnh cht n iu ca hm s v kt lun.V d 1: Tm tham s thc m ptrnh 23 1x x m+ + = c nghim thc .

Gii :

*Xt hm s ( ) 23 1 f x x x = + + v y m= *Hm s ( ) 23 1 f x x x = + + lin tc trn .* Ta c : ( ) 2

2 2

3 3 1 3' 1

3 1 3 1

x x xf x

x x

+ += + =

+ +

( ) 2 2 20

' 0 3 1 33 1 9

x f x x x

x x

nghch bin trn na khong )0; + v


44/45

Nguyn Ph Khnh Lt .

46

lim ( ) 0x

f x +

= , nn )0 ( ) 1, 0; f x x < + .

Vy, phng trnh ( )1 c nghim thc trn na khong )0; 0 1m + <

V d 3: Tm tham s thc m phng trnh :

( ) ( ) ( )4 3 3 3 4 1 1 0, 2m x m x m + + + = c nghim thc.Gii :

iu kin: 3 1x .

* Phng trnh 3 3 4 1 1(2)4 3 3 1 1

x xm

x x

+ + + =

+ + +

*Nhn thy rng:( ) ( )

2 22 2 3 1

3 1 4 12 2

x xx x

+ + + = + =

*Nn tn ti gc 0; , t n ; 0;12 2

t a t

=

sao cho:

2

23 2 sin 2

1

tx

t+ = =

+v

2

2

11 2 cos 2

1

tx

t

= =

+

( ) ( )2

2

3 3 4 1 1 7 12 9, 3

5 16 74 3 3 1 1

x x t t m m f t

t tx x

+ + + + += = =

+ ++ + +

*Xt hm s: 22

7 12 9( )

5 16 7

t tf t

t t

+ +=

+ +

lin tc trn on 0;1t

. Ta c

( )( )

2

22

52 8 60'( ) 0, 0;1

5 16 7

t t f t t f t

t t

= <

+ +

nghch bin trn on [ ]0;1

v9 7

(0) ; (1)7 9

f f= =

* Suy ra phng trnh ( )2 c nghim khi phng trnh ( )3 c nghim trnon 0;1t khi v ch khi:

7 9

9 7m .

V d 4: Tm tham s thc m bt phng trnh2 22 24 2x x x x m + + c nghim thc trong on 4;6 .

Gii :


45/45

Nguyn Ph Khnh Lt .

t 2 2 24t x x= + , 4;6 0;5x t

Bi ton trthnh tm tham s thc m bt phng trnh 2 24t t m+ c

nghim thc 0;5t

*Xt hm s ( ) 2 24 f t t t = + lin tc trn on 0;5 .* Ta c : ( )'( ) 2 1 0, 0;5 f t t t f t = + > lin tc v ng bin trnon 0;5

*Vy bt phng trnh cho c nghim thc trn on 0;5 khi0;5

max ( ) (5) 6 6t

f t m f m m m

Bi tp tluyn:

1. Tm tham s thc m phng trnh : ( )1 2 1mx m x + + = c nghim

thc trn on 0;1 .

2. Tm tham s thc m bt phng trnh: 2 24 5 4x x x x m + + c

nghim thc trn on 2;3 .

Dng 8 : Dng n iu hm s chng minh h thc lng gic

V d : Chng minh rng : nu tam gic ABC tho mn h thc1 13

cos cos coscos cos cos 6

A B CA B C

+ + + =+ +

th tam gic ABC u.

Gii :

*t : 3cos cos cos 1 4 sin sin sin 12 2 2 2

A B Ct A B C t = + + = + < .

*Xt hm s ( ) 1 f t t t

= + hm s lin tc trn na khong3

0;2

.

* Ta c : ( ) ( )21 3' 1 0, 0; 2 f t t f t t

= >

ng bin trn na khong

30;

2

,suy ra : ( )13

26

f t< .

ng thc ( ) 136

f t = xy ra khi 3cos cos cos2

t A B C = + + = hay tam gic

ABC u.

Chuong[1]-Bai[1] - Ung Dung Dao Ham Khao Sat - Ve Do Thi Ham So

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Transcript of Chuong[1]-Bai[1] - Ung Dung Dao Ham Khao Sat - Ve Do Thi Ham So