Định Lý Lagrange, Rolle, Cauchy Và Ứng Dụng

5/21/2018 nh L Lagrange, Rolle, Cauchy V ng D ng

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LI CM N

Kha lun ny c hon thnh di s gip nhit tnh, chu o ca Th.S PhanTrng Tin. Em xin php c gi n thy s knh trng v lng bit n su sc v

s tn tm ca thy i vi bn thn em, khng nhng trong qu trnh lm kha lun

m cn trong sut qu trnh hc tp.

Em cng xin php c gi li cm n chn thnh n qu thy c ging dy

lp HSP Ton L K51, cng nh ton th thy c gio trong khoa Ton Tin,

trng H Qung Bnh, nhng ngi cho em kin thc, quantm, ng vin, nhit

tnh gip em trong sut qu trnh hc tp cng nh trong thi gian thc hin khalun.

Cui cng, em xin php c gi li cm n n nhng ngi thn, bn b

quan tm, ng vin, gip em trong sut qung ng hc tp va qua.

ng Hi, thng 5 nm 2013

Sinh vin: on Th Tr My


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MC LC

A. M U............................................................................................................... 1

I. L DO CHN TI.................................................................. ................. 1

II. MC CH NGHIN CU................................................................. .......... 1

III.PHM VI NGHIN CU I TNG NGHIN CU......................... 1

IV.NI DUNG NGHIN CU................................................................. ........... 2

V. CC PHNG PHP NGHIN CU V TIN HNH............................ 2

B. NI DUNG............................................................................................................ 3

CHNG I_KIN THC CHUN B............................................. ....................... 3

I. GII HN CA HM S................................................. .............................. 3

II. HM S LIN TC........................................................................................ 4

III. O HM ....................................................................................................... 6

CHNG II_NG DNG CC NH L BOLZANO-CAUCHY,LAGRANGE,ROLLE, CAUCHY CHNG MINH PHNG TRNH C NGHIM.......... 9

I. NG DNG NH L BOLZANO-CAUCHY ............................................. 9

1. Phng php chung................................................................ ................. 9

2. Bi tp...................................................................................................... 9

II. NG DNG NH L LAGRANGE, ROLLE, CAUCHY ......................... 26

1. Phng php chung................................................................ ............... 262. Bi tp.................................................................................................... 28

CHNG III_NG DNG NH L LAGRANGE GII PHNG TRNH.. 81

I. PHNG PHP CHUNG................................................ ............................ 81

II. BI TP.................................................. .............................................. ....... 81

CHNG IV_NG DNG NH L LAGRANGE CHNG MINHBTNG THC ........................................................................................................ 90

I. PHNG PHP CHUNG................................................ ............................ 90II. BI TP.................................................. .............................................. ....... 90

C. KT LUN....................................................................................................... 109

TI LIU THAM KHO..................................................................................... 110


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1

A.M U

I.

L DO CHN TIo hm l mt phn quan trng ca gii tch ton hc. N khng nhng l mt

i tng nghin cu ca gii tch, m cn l cng c rt "mnh" gii quyt hu ht

nhng bi ton trong cc thi tt nghip Trung hc ph thng cng nh trong cc

thi tuyn sinh i hc, Cao ng.

Trong nhng nm gn y, nhng k thi hc sinh gii cp quc gia, quc t,trong

cc k thi Olympic Ton Sinh vin gia cc trng i hc trong nc th cc bi ton

lin quan n tnh lin tc v o hm ca hm shay c cp, v c xem nh l

dng ton kh. Dng ph bin nht l chng minh phng trnh c nghim, gii phng

trnh, chng minh bt ng thc.

Cc bi ton ny hu ht c cp cc cun bi tp gii tch. Tuy nhin, ti

liu h thng v ng dng ca cc nh l v tnh lin tc v o hm gii cc bi

ton ny th cha c nhiu, cn cha c h thng theo dng ton cng nh phng

php gii. Qua nghin cu k ni dung kin thc, c nhiu ti liu, qua kinh nghimhc Ton ca bn thn, em chn ti Cc nh l Bolzano Cauchy, Lagrange,

Rolle, Cauchy v ng dng.Nhm gip cho cc em hc sinh, sinh vin c thm ti liu

tham kho, gip cho vic hc Ton thm phn hp dn hn.

II. MC CH NGHIN CU

Nghin cu cc ng dng ca cc nh l Bolzano Cauchy, Lagrange, Rolle,

Cauchy chng minh phng trnh c nghim, gii phng trnh v chng minh bt

ng thc.

III.PHM VI NGHIN CU I TNG NGHIN CU

i tng nghin cu ca ti ch yul cc bi tp ra trong cc sch gii tch,

cc thi Olympic lin quan n ng dng lin tc v o hm.


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2

IV.

NI DUNG NGHIN CU

CHNG I_KIN THC CHUN B

CHNG II_NG DNG CC NH L BOLZANO-CAUCHY,

LAGRANGE, ROLLE, CAUCHY CHNG MINH PHNG TRNH C

NGHIM

CHNG III_NG DNG NH L LAGRANGE GII PHNG TRNH

CHNG IV_NG DNG NH L LAGRANGE CHNG MINH

BT NG THC

V.CC PHNG PHP NGHIN CU V TIN HNH

-Tham kho ti liu.

-H thng cc bi tp v phn loi.

-Phn tch, hng dn phng phpgii.


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3

B. NI DUNG

CHNG I_KIN THC CHUN B

I.

GII HN CA HM S

1.

Cc nh ngha

Cho hm s y f x xc nh trn khong ,a b cha ox (c th khng xc nh

ti imo

x ).

nh ngha 1.1:S thc l c gi l gii hn ca hm s y f x khi x dn n

ox nu 0, 0 : ,x a b m 0 ox x f x l .

K hiu: limox xf x l

hay f x l khi ox x .

nh ngha 1.2: S L c gi l gii hn phi (gii hn tri) ca hm f x khi x

tin tio

x t bn phi (t bn tri) nu vi mi 0 tn ti s 0 sao cho

f x L vi mi , ,0 (0 )o ox a b x x x x .

K hiu:Gii hn phi: lim ox a

f x f x

hoc 0of x ;

Gii hn tri: lim ox a

f x f x

hoc 0of x ;

nh ngha 1.3:Nu vi mi s 0M , tn ti s 0 sao cho:

( )f x M f x M vi mi x tha mn bt ng thc 0 ox x th ta ni

f x c gii hn bng ( ) khi x tin ti ox .

K hiu: lim (lim )o ox x x xf x f x

.

Gi s hm

f x xc nh trn tp khng b chn.

nh ngha 1.4:S L c gi l gii hn ca f x khi x tin ra ( ) nu

vi mi 0 tn ti s 0M sao cho vi mi ,x a b tha mn bt ng thc

x M x M ta c: f x L .

K hiu: lim ( lim )x x

f x L f x L

.


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nh ngha 1.5: Nu vi mi s 0E tn ti s 0M sao cho

( )f x E f x E vi mi x X tha mn x M th ta ni hm f x c gii

hn ( ) khi x tin ra .

K hiu: lim ( lim )x x

f x f x

;

Tng t cho lim ( lim )x x

f x f x

.

2.Cc nh l

nh l 1.6:Gi s f x l mt hm n iu trn khong ,a b v c l mt im

nm trong khong . Nu f x b chn th tn ti gii hn tng pha (hu hn)

limx c

f x

v limx c

f x

.

nh l 1.7: Gi s tn ti 0 sao cho vi mi ,x a b tha mn

0 x a , hm f x b kp gia hai hm ,g x h x (tc l

g x f x h x ) v tn ti lim limx a x a

h x g x L

. Khi y tn ti gii hn ca

f x khi x tin ti a v limx a

f x L

.

II.

HM S LIN TC

1.

Cc nh ngha

Gi s hm ( )f x xc nh trn mt ln cn ca im ox .

nh ngha 2.1: Hm ( )f x c gi l lin tc ti im ox nu vi mi 0 tn

ti mt s 0 sao cho vi mi x : ox x ( ) ( )of x f x .

Ta ni ( )f x gin on ti ox nu n khng lin tc ti im .

Tuy nhin ta c th nh ngha yu hn:

nh ngha 2.2:Hm ( )f x c gi l lin tc bn tri ti imo

x nu n xc nh

trong ( , ]o o

x x v lim ( ) ( )o

ox x

f x f x

.

Hm ( )f x c gi l lin tc bn phi ti im ox nu n xc nh trong

[ , )o o

x x v lim ( ) ( )o

ox x

f x f x

.


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Hm ( )f x lin tc tio

x nu lin tc bn tri v lin tc bn phi.

nh ngha 2.3: Hm s ( )f x c gi l lin tc Lipschitz nu c hng s A

, ,f x f y A x y x y X .

2.

Cc nh lnh l 2.1: Nu hm ( )f x v ( )g x lin tc ti im

ox x th cc hm

( )( ) ( ), ( ). ( ), ( ( ) 0)

( ) of x

f x g x f x g x g xg x

lin tc ti im ox x .

nh l 2.2:Nu hm ( )t g x lin tc ti imo

x , hm ( )y f t lin tc ti im

0( )ot g x th hm hp [ ( )]y f g x lin tc ti im ox .

nh l 2.3:N

u hm ( )

f xxc nh v

lin tc ti im

ox

th n b chn trong

mt ln cn no ca imo

x .

nh l 2.4(Weierstrass I):Nu hm ( )f x xc nh v lin tc trn on ,a b th

n b chn trn on .

nh l 2.5(Weierstrass II):Nu hm ( )f x lin tc trn on ,a b th n t gi

tr ln nht v gi tr nh nht trn on . Tc l tn ti cc im 1 2, ,x x a b sao

cho:

1,

maxx a b

f x M f x

.

2,

minx a b

f x m f x

.

nh l 2.6(Bolzano Cauchy I):Nu ( )f x lin tc trn ,a b v ( ). ( ) 0f a f b

th c t nht mt im ,c a b sao cho 0f c .

nh l 2.7 (Bolzano Cauchy II): Gi s ( )f x lin tc trn ,a b v

( ) ( )f a A B f b .

Khi y ( )f x nhn mi gi tr trung gian gia A v B. (Ta ni: ( )f x lp y on

,A B ).


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H qu:Nu hm s ( )f x lin tc trn ,a b th n nhn mi gi tr trung gian

gia gi tr ln nht v gi tr nh nht.

nh l 2.8:Nu ( )f x lin tc Lipschitz th ( )f x lin tc u.

III.O HM

1.

Cc nh ngha

Gi s hm ( )f x xc nh trn khong ,a b v x l im c nh ca khong ,

x l s gia ty ca i s b sao cho s x x cng thuc khong ,a b .

y l s gia ca hm ( )y f x ti im c nh .

nh ngha 3.1:Nu tn ti gii hn hu hn

0 0

( ) ( )lim limx x

y f x x f x

x x

(1)

Th gii hn c gi l o hm ca hm ( )y f x ti im c nh x .

Ta ni f x c o hm trn ,a b ( hay kh vi trn ,a b ) nu f x c o hm

ti mi im thuc ,a b ).

Trong trng hp gii hn (1) tn ti v bng hoc th ngi ta ni hm

( )f x c o hm v hn ti ox .

nh ngha 3.2:Nu cc gii hn sau y tn ti:

0 0

( ) ( )lim limx x

y f x x f x

x x

.

0 0

( ) ( )lim limx x

y f x x f x

x x

.

Th cc gii hn c gi tng ng l o hm bn phi v o hm bn tri

ca hm ( )y f x ti im c nh x .

K hiu:

0 0'( 0) lim

x

yf x

x

l o hm bn phi.


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0 0'( 0) lim

x

yf x

x

l o hm bn tri.

T cc nh ngha 3.1, 3.2v cc tnh cht ca gii hn mt pha ta thu c: o

hm '( )f x tn ti khi v ch khi ti im x hm ( )f x c cc o hm bn phi v bn

tri v cc o hm bng nhau:

'( 0) '( 0) '( )f x f x f x .

2.

Cc nh l Lagrange, Rolle, Cauchy.

a.nh l Lagrange:

Gi s ( )f x l hm s lin tc trn on ,a b v c o hm trn ,a b . Khi

tn ti mt im ,c a b sao cho:

( ) ( )'( )

f b f af c

b a

.

* nh l Lagrange cn c pht biu di dng tch phn sau:

Nu ( )f x l hm s lin tc trn on ,a b th tn ti im ,c a b tha

mn: b

a

f x dx f c b a .

* ngha hnh hc: Cho hm s f x tha mn cc gi thit ca nh l

Lagrange. th C , ,A a f a , ,B b f b .

Khi trn C tn ti im

,C c f c , ,c a b l tip tuyn ca C ti

Csong song vi ng thng AB .

nh l Lagrange cho php ta c lng t

s f b f a

b a

do n cn c gi l

nh l Gi tr trung bnh (Mean Value Theorem).

AC

B

ba cO

y

x


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b.

nh l Rolle:

Nu hm ( )f x lin tc trn on ,a b , kh vi trn khong ,a b v ( ) ( )f a f b

th tn ti ,c a b sao cho '( ) 0f c .

nh l Rolle l h qu ca nh l Lagrange trong trng hp f a f b

.

c.nh l Cauchy:

Gi s cc hm ( )f x v ( )g x l cc hm lin tc trn ,a b , kh vi trn ,a b .

Khi tn ti ,c a b cho ( ) ( ) '( ) ( ) ( ) '( )f b f a g c g b g a f c .

nh l Lagrange l h qu ca nh l Cauchy trong trng hp g x x .


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9

CHNG II_NG DNG CC NH L BOLZANO-CAUCHY,

LAGRANGE, ROLLE, CAUCHY CHNG MINH PHNG TRNH C

NGHIM

I. NG DNG NH L BOLZANO-CAUCHY

1.

Phng php chung

Cho phng trnh ( ) 0f x , chng minh phng trnh c k nghim phn bit

trong ,a b , ta thc hin theo cc bc sau:

Bc 1: Chn cc s 1 2 1... ka T T T b chia on ,a b thnh cc khong

tha mn:

1

1

( ). ( ) 0

........

( ). ( ) 0k

f a f T

f T f b

Bc 2:Kt lun.

2.Bi tp

Bi ton 1:Cho cc s dng 1 2 3, ,c c c tha mn 1 2 3c c c . Chng minh rng

phng trnh 1 2 3x c x c x c c nghim duy nht .

* Phn tch: chng minh phng trnh 0f x c nghim duy nht:

+ Chng minh 0f x c nghim ox trong tp xc nh ca n: Da vo tnh lin

tc ca hm s f x trn tp xc nh v ch ra 1 2. 0f x f x , 1 2,x x D (S dng

nh l Bolzano-Cauchy).

+ Bng cng c o hm, chng t f x n iu trn min D .

T , suy ra ox l nghim duy nht.

* Gii:

iu kin : 1x c .

Khi phng trnh c a v dng : 1 2

3 3

1 0x c x c

x c x c

.


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Xt hm s 1 23 3

1x c x c

f xx c x c

trn 1;c .

Ta c

1 3 2 3

2 21 23 3

3 3

0

2 2

c c c cf x

x c x cx c x c

x c x c

.

Do gi thit 1 2 3c c c nn hm s f x ng bin trn 1;c .

Mt khc 1 211 3

1 0c c

f cc c

v lim 1

xf x

.

S dng nh l Bolzano - Cauchy ta thy phng trnh 0f x c duy nht

nghim 0 1; .x c

Bi ton 2:Chng minh phng trnh: 3 3 1 0x x c 3 nghim phn bit. Tnh

tng cc lu tha bc 8 ca 3 nghim .

* Phn tch: chng minh phng trnh c 3 nghim ta s ch ra trn tp xc nh

ca phng trnh c 3 khong , , , , ,a b b c c d sao cho:

. 0, . 0, . 0.f a f b f b f c f c f d

Ri vn dng nh l Bolzano Cauchy ta c iu cn chng minh.

* Gii:

- Hm s 3 3 1f x x x lin tc trn D .

Xt hm s f x trong cc khong 2,0 ; 0,1 ; 1, 2 .

Ta c :

- Trong 2,0 hm s f x lin tc v: 3

2 2 3 2 1 1 0f ;

0 1 0f .

Suy ra: 2 . 0 0f f .

p dng nh l Bolzano Cauchy: 1 2,0x sao cho 1 0f x .

- Trong 0,1 hm s f x lin tc v:

0 1 0f ; 3

1 1 3. 1 1 2 0f .


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Suy ra: 0 . 1 0f f .


- Trong 1,2 hm s f x lin tc v: 3

1 1 3. 1 1 2 0f ;

3

2 2 3.2 1 1 0f

.Suy ra: 1 . 2 0f f .


Nn phng trnh cho c 3 nghim phn bit l 1 2 3, ,x x x (pcm).

* Tnh tng cc lu tha bc 8 ca 3 nghim tc l tnh gi tr ca 8 8 81 2 3x x x .

Gi , 1,3i

x i l nghim ca phng trnh cho .

Ta c: 3 33 1 0 3 1i i i ix x x x .

Ngoi ra:

+ 5 3 2 2 3 2 2 2. 3 1 . 3 3 3 1 9 3i i i i i i i i i i ix x x x x x x x x x x . (1)

+ 8 5 3 2 2. 9 3 3 1 28 27 6i i i i i i i ix x x x x x x x . (2)

Do : 8 8 8 2 2 21 2 3 1 1 2 2 3 328 27 6 28 27 6 28 27 6x x x x x x x x x

2 2 2

1 2 3 1 2 328 27 18x x x x x x

.

Mt khc: 22 2 2

1 2 3 1 2 3 1 2 2 3 3 12x x x x x x x x x x x x .

Nn3

8

1i

i

T x

2

1 2 3 1 2 2 3 3 1 1 2 319 2 27 18x x x x x x x x x x x x .

V phng trnh cho c 3 nghim phn bit nn theo nh l Viet:

1 2 3

1 2 2 3 3 1

1 2 3

0

. . . 3

. . 1

x x x

x x x x x x

x x x

Vy3

8

1

28.6 6 186i

i

T x

.


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Bi ton 3:Chng minh tp nghim ca bt phng trnh:

1 2 70 5...

1 2 70 4x x x

l hp cc khong ri nhau v c tng di l 1988.

* Phn tch: Bi ton cn chng minh tp nghim ca bt phng trnh cho lhp cc khong ri nhau. Tc l chng minh bt phng trnh c cc nghim 1 2, ,... nx x x

m cc nghim thuc vo nhng khong khc nhau:

1 1 2 2 2 3 1, , , ,.... ,n n nx a a x a a x a a .

Ngoi ra:70

1

1 2 70 5 5...

1 2 70 4 4k

k

x x x x k

70 70

1 1

( ) 4 5

5( ) 4 4 ( )

k j k j k

k x j k x j x j

x j x j

( )( )

f xg x

.

Ta thy hm s f x lin tc trn nn chng minh s tn ti nghim trong

cc khong ca bt phng trnh trn, ta c ths dng nh l Bolzano-Cauchy.

* Gii:

+ Ta c:70

1

1 2 70 5 5...

1 2 70 4 4k

k

x x x x k

70 70

1 1

( ) 4 55

( ) 4 4 ( )k j k j k

k x j k x j x j

x j x j

( )

( )

f x

g x .

Vi qui c , 1,70k j .

- R rng ( ) 0g x c 70 nghim 1,2,...70x .

- Ta c f x lin tc trn .

V ( ). ( 1) 0, 1,69f k f k k .Nn theo nh l Bolzano-Cauchy c 69 nghim xen k gia cc nghim

1,2,...70x .

Ngoi ra: lim ( ) 0, (70) 0x

f x f

.

Nn cng c 70 nghim xen k l: 1 2 69 701 2 ... 70x x x x .


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Nh vy,tp nghim ca bt phng trnh cho l hp cckhong ri nhau.

+ Tng di cc khong nghim ca bt phng trnh:( )

0( )

f x

g x l:

1 2 70

1 2 70

1 2 ... 70

... (1 2 ... 70)

S x x x

x x x

a thc f x c bc 70, h s cao nht l 5 v h s ca 69x l:

9 1 2 ... 70 .

Do :

9 1 2 ... 70 4 70.71

1 2 ... 70 . 19885 5 2

S

.

Bi ton 4: Cho hm s f x : ; ;a b a b vi a b v tho mn iu kin:

f x f y x y ,vi mi ,x y phn bit thuc ;a b .Chng minh rng phng trnh f x x c duy nht mt nghim thuc ;a b .

(Olympic sinh vin 1994)

* Phn tch: Hm s f x tha mn iu kin cho l mt hm lin tc

Lipschitz (Theo nh ngha hm lin tc Lipschitz). Tuy nhin ta cha th khng nh

c n l hm kh vi.

Thng thng, chng minh phng trnh 0f x c nghim duy nht, ta s

da vo nh l Bolzano-Cauchy chng minh n c nghimox trn tp xc nh. Ri

bng cng c o hm chng t f x n iu trn min D . T , suy ra ox l

nghim duy nht.

Tuy nhin, bi ton ny hm s khng c xt trn mt on c th. Nn ta

khng th s dng nh l Bolzano-Cauchy trc tip, m ta s dng nh l Weierstrass

chng minh 0f x c nghim ox , ri bng phn chng chng minh ox l duy

nht.

* Gii:

- Xt hm s g x f x x th ( )g x lin tc trn ;a b .

Do tn tio

x thuc ;a b sao cho: 0 ,( ) min ( )

x a bg x g x

. (*)


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- Ta s chng minh 0og x .

Tht vy, gi s 0og x , do o of x x .

T bt ng thc cho th c: o o o of f x f x f x x .

Suy ra o og f x g x : mu thun vi (*).

Vy 0og x ngha l o of x x .

- Gi s phng trnh f x x cn c nghim 1 ox x , 1 ;x a b th ta c:

1 1,o of x x f x x 1 1o of x f x x x (mu thun vi gi thit).

V theo gi thit, nu x y th f x f y x y iu gi s sai.

Vy, phng trnh f x x c duy nht mt nghim thuc ;a b .

Bi ton 5:Cho , , 0a b c v ,p q ty . Chng minh rng:

2 2a bc

x p x q

lun c nghim.

* Phn tch:V ,p q ty , ta xt hai trng hp p q v p q .

Ta thy: Vi p q : hin nhin phng trnh lun c nghim.

Vi p q , gi s p q trong iu kin l ,x p x q , quy ng kh mu

phng trnh cho, ri p dng nh l Bolzano-Cauchy cho hm s l v tri ca

phng trnh trn ,p q ta s c c iu phi chng minh.

* Gii:

+ Vi p q

ta c

2 2

2 2

x pa b

c a bx p x pc

Vy, phng trnh c nghim2 2a b

x pc

khi p q v x p .

+ Vi p q , gi s p q iu kin xc nh ,x p x q .


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Vi iu kin phng trnh cho tng ng vi:

2 2

2 2

( ) ( ) ( )( )

( )( ) ( ) ( ) 0

a x q b x p c x p x q

c x p x q a x q b x p

t 2 2( )f x c x p x q a x q b x p .

Ta c ( )f x lin tc trn ,p q v 2 2,f p a p q f q b q p .

Suy ra: 2 2 2( ). ( ) . .( ) 0f p f q a b p q .

Do , tn ti so

x gia ,p q sao cho ( ) 0o

f x .

Tc l phng trnh2 2a b

cx p x q

lun c nghim (pcm).

Bi ton 6: Cho hm s ( )f x lin tc trn on 0,1 tha mn iu kin

0 1f f .

Chng minh rng phng trnh: 1

2000f x f x

c nghim 0,1x .

* Phn tch:

Bi ton cn chng minh phng trnh:

1 1

02000 2000

f x f x f x f x

c nghim 0,1x .

t 1

2000g x f x f x

.

Ta thy rng hm s g x lin tc trn 0,1 , v cha xc nh c hm s g x

kh vi hay khng nn ta c th s dng nh l Bolzano-Cauchy chng minh rng

0,1x sao cho 0g x .

* Gii:

- Xt hm s 1

2000g x f x f x

.


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Hm s ny xc nh v lin tc trn1999

0,2000

.

Ta c:

10 0

2000

1 2 1

2000 2000 2000

2 3 2

2000 2000 2000

.......................................................

1999 19991

2000 2000

g f f

g f f

g f f

g f f

Suy ra: 1 1999

0 ... 1 0 02000 2000g g g f f

.

T suy ra tn ti , 0,1,...,1999i j ta c: 0i

gn

v 0

jg

n

.

+ Nu 0i

gn

v 0

jg

n

th ta c iu phi chng minh.

+ Nu 0i

g

n

v 0j

g

n

th do ( )g x lin tc trn1999

0,

2000

nn theo nh l

Bolzano-Cauchy tn ti ,oi j

xn n

sao cho phng trnh ( ) 0g x c nghim trn

19990,

2000

.

Hay phng trnh 1

2000f x f x

c nghim thuc on 0,1 .

Vy, phng trnh 1

2000f x f x

lun c nghim 0,1x (pcm).


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17

Tng qut:Cho f x l mt hm s lin tc trn on 0,1 tha mn iu kin

0 1f f . Chng minh rng vi bt k s t nhin n no cng tn ti mt s

0,1c sao cho: 1cn

f c fn

.

Hng dn: Xt hm s 1 1

, 0,n

g x f x f x xn n

.

Bi ton 7: Chng minh rng phng trnh:

0p x a x c q x b x d lun c nghim, bit rng a b c d , ,p q

l hai s thc ty .

* Phn tch:t f x p x a x c q x b x d .

V ,p q l hai s thc ty nn ta sxt hai trng hp l:

+ 0p q : R rng phng trnh lun c nghim.

+ 0p (hoc 0q ):

Nhn xt: f x lin tc trn ,b d nn ta s p dng nh l Bolzano-Cauchy

chng minh s tn ti nghim ca phng trnh cho (Tng t, ta cng c th xt

hm s f x trn on ,a c)

* Gii:

Xt hm s f x p x a x c q x b x d lin tc trn .

+ Nu 0p q , phng trnh c nghim ty .

+ Nu 0p hoc 0q , khng mt tnh tng qut ta gi s 0p .

Khi : f b p b a b c

.

V f d p d a d c .

Suy ra: 2. 0f b f d p b a b c d a d c .

Vy phng trnh lun lun c nghim.


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18

Bi ton 8:Chng minh rng: Vi mi m phng trnh

a/1 1

cos sin m

x x lun lun c nghim.

b/ .sin3 6cos 2 sin 0m x x x c nghim 0,2x .

* Phn tch:

a/ t iu kin cho bi ton. Thc hin cc php bin i a cc s hng v

mt v sin cos .sin .cos 0x x m x x .

t sin cos .sin .cosf x x x m x x .

chng minh phng trnh lun c nghim, ta ch cn ch ra mtkhong m trn

hm s f x lin tc, ri s dng nh l Bolzano-Cauchy c iu phi chng

minh.

b/ Ta nhn thy hm s .sin3 6cos2 sinf x m x x x lin tc trn 0,2 .

Cng tng t nh cu trn chng minh phng trnh

.sin3 6cos2 sin 0m x x x c nghim 0,2x . Ta cng s dng nh l Bolzano-

Cauchy.

* Gii:

a/ iu kin2

x k

, vi *k .

Bin i phng trnh v dng: sin cos .sin .cos 0x x m x x .

Xt hm s sin cos .sin .cosf x x x m x x lin tc trn on 0,2

.

Ta c: 0 1 0f v 1 02f

.

Suy ra: 0 . 1 02

f f

.

Vy phng trnh 0f x lun c mt nghim thuc 0,2

.


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19

Hay phng trnh cho lun c nghim.

b/ Xt cos3 cos 2 cos sinf x a x b x c x x th ( )f x lin tc trn 0, 2

.

Ta c:

+ Ti 0x : 0f a b c .

+ Ti2

x : 1

2f b

.

+ Ti x : f a b c .

+ Ti3

2x

:

31

2f b

.

Suy ra: 3

0 02 2

f f f f

.

Do 3

, 0, , ,2 2

. 0f f .

Vy phng trnh c nghim3

0,2

x

hay c nghim 0,2x .

Bi ton 9:

Cho f x lin tc trn on 0,1 , 0 0f v1

0

1( ) , 0

1998f x dx x .

Chng minh rng: 1997 ( )x f x lun c t nht mt nghim thuc 0,1 .

(Olympic sinh vin 1998)

* Phn tch:

Bi ton cn chng minh phng trnh 1997 1997( ) ( ) 0x f x x f x lun c t

nht mt nghim thuc 0,1 .


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20

Theo bi ra: f x lin tc trn on 0,1 , ngoi ra cn c 0 0f v

1

0

1( )

1998f x dx nn ta s ngh ti vic s dng nh l Bolzano Cauchy chng

minh phng trnh trn c nghim.

t 1997( ) ( )F x x f x , ri xt hm s ( )F x trn 0,1 , ta s c iu phi chng

minh.

* Gii:

Xt hm s 1997( ) ( )F x x f x .

Khi ( )F x lin tc trn 0,1 v theo gi thit th 0 0F .

V 1 1

0 0

1 01998

F x dx f x dx .

Suy ra 1 0,1x sao cho 1 0F x .

Do ( )F x lin tc trn 10,x v 10 . 0F F x nn 10,c x 0F c .

Vy phng trnh 1997 ( )x f x lun c t nht mt nghim thuc 0,1 (pcm).

Bi ton 10:Cho s thc 2a v

10 10 1 ... 1n n nn

f x a x x x x .

Chng minh rng vi mi s nguyn dng n , phng trnh nf x a lun c

ng mt nghim dng duy nht. K hiu nghim ln

x .

(VMO 2007)

* Phn tch:

Bi ton cn chng minh phng trnh 0n nf x a f x a lun c ng

mt nghim dng duy nht. Vi 10 10 1 ... 1n n nnf x a x x x x v 2a .

+ Ta s chng minh phng trnh 0nf x a c nghim: V hm s nf x lin

tc trn , ng thi bi ton khng cho thm d kin no, nn ta ss dng nh l

Bolzano Cauchy chng minh s tn ti nghim ca phng trnh trn.


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21

t n nF x f x a .

+ Bng cng c o hm, chng t nF x n iu trn min D .

T , suy ra phng trnh 0nF x ch c nghim duy nht.

* Gii:t n nF x f x a .

Ta c: nF x lin tc trn 0, .

10 9 1 2' 10 1 ... 2 1 0n n nnF x n a x nx n x x .

Vy, nF x ng bin trn 0, .

Ngoi ra, ti 0x : 0 1 0nF a .

Ti 1x : 100 1 0nF a n .

Theo nh l Bolzano Cauchy, 0,1nx sao cho:

0 0n n n nF x f x a f x a .

Vy, 0,1nx sao cho nf x a .

Hay phng trnh nf x a lun c mt nghim dng nx duy nht(pcm).

Bi ton 11: Cho hai hm f x v g x lin tc trn 0,1 v tha mn:

0 1 0, 1 0 1f g f g .

Chng minh rng, 0, 0,1x f x g x .

* Phn tch:Bi ton cn chng minh 0,1x f x g x .

T kt qu 0f x g x f x g x , ta t hm tng ng l

h x f x g x .

Nhn xt:Theo bi ra f x v g x lin tc trn 0,1 nn hm s h x cng lin

tc trn 0,1 .

V vy chng minh s tn ti nghim th ta s s dng nh l Bolzano-Cauchy.


24/112

22

* Gii:

- Xt hm s: h x f x g x .

Ta thy hm s h x lin tc trn 0,1 .

Ngoi ra, 0 0 0 0 0h f g v 1 1 1 1 0h f g .

Suy ra: 0 . 1 0h h .

Theo nh l Bolzano-Cauchy, 0,1x sao cho 0h x f x g x .

Bi ton 12: Chng minh rng tn ti s thc 0,1x sao cho:

1 2000 2001

2 2001 2 2001

1 1 ... 1 1 1 ... 1x

t dt x

t t t x x x

* Phn tch:Bi ton cn chng minh 0,1x

1 2000 2001

2 2001 2 20011 1 ... 1 1 1 ... 1x

t dt x

t t t x x x

T kt qu:

1 2000 2001

2 2001 2 20011 1 ... 1 1 1 ... 1x

t dt x

t t t x x x

1 2000 2001

2 2001 2 20010

1 1 ... 1 1 1 ... 1x

t dt x

t t t x x x

.

Ta t

1 2000 2001

2 2001 2 20011 1 ... 1 1 1 ... 1x

t dt xf x

t t t x x x

.

Nhn xt:Hm s f x lin tc trn 0,1 .

V vy, chng minh s tn ti nghim th ta c th s dng nh l Bolzano Cauchy.

* Gii:

Xt hm s:

1 2000 2001

2 2001 2 20011 1 ... 1 1 1 ... 1x

t dt xf x

t t t x x x

.


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23

R rng, f x lin tc trn 0,1 .

Ta c:

1 2000

2 20010

0 01 1 ... 1

t dtf

t t t.

200111 02f .

Suy ra: 0 . 1 0f f .

Theo nh l Bolzano-Cauchy: 0,1x sao cho 0f x hay

1 2000 2001

2 2001 2 20011 1 ... 1 1 1 ... 1x

t dt x

t t t x x x

( pcm).

Bi ton 13:Cho phng trnh:1

1 .n

i

ni nx

Chng minh rng: Vi mi s nguyn dng n phng trnh c duy nht mt

nghim dng. K hiu nghim lnx .

* Phn tch:Bi ton yu cu chng minh phng trnh:

1 1

1 10

n n

i i

n ni nx i nx

c duy nht mt nghim dngn

x .

+ u tin ta s chng minh phng trnh c nghimn

x trong [0; ) .

V bi ton khng cho thm d kin no khc nn ta s s dng nh l Bolzano-

Cauchy chng minh.

+ Tip theo, ta s chng minh phng trnh c nghimduy nht bng cng c o

hm.

* Gii:

Xt1

1( )

n

i

f x ni nx

.

Ta c:3

1

1'( ) 0, (0; )

2 ( )

n

i

f x xi nx

.


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24

Suy ra f xlin tc, nghch bin trn [0; ) .

M1

1(0) 0

n

i

nf n n

i n .

lim ( ) 0x

f x n

.

Vyphng trnh ( ) 0f x c 1 nghim dng duy nht(pcm).

Bi ton 14:Xt phng trnh2 2

1 1 1 1 1... ...

1 4 1 1 1 2x x k x n x

(1)

Trong n l tham s dng.

Chng minh rng vi mi s nguyn dng n , phng trnh nu trn c duy nht

nghim ln hn 1, k hiu nghim ln

x . (VMO 2002)

* Phn tch:Bi ton yu cu chng minh phng trnh:

2 2

1 1 1 1 1... ...

1 4 1 1 1 2x x k x n x

c duy nht nghim

nx ln hn 1.

Ta c:

2 2

1 1 1 1 1... ...

1 4 1 1 1 2x x k x n x

2 2

1 1 1 1 1... ... 0

2 1 4 1 1 1x x k x n x

.

+ u tin, chng minh phng trnh cho c nghim trong 1, :

t 2 21 1 1 1 1

... ...2 1 4 1 1 1n

f xx x k x n x

.

Ta thy, hm s nf x lin tc, v khng c thm d kin no trong bi nn ta s

s dng nh l Bolzano Cauchy chng minh phng trnh c nghim.

+ S dng cng c o hm chng minh nf x n iu trn 1, .

T y, ta kt lun c phng trnh cho c duy nht nghim ln hn 1.


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* Gii:

Xt hm s 2 21 1 1 1 1

... ...2 1 4 1 1 1n

f xx x k x n x

.

Vi mi *n , hm s nf x lin tc v nghch bin trn khong 1, .

Ngoi ra: 1

lim nx

f x

.

1

lim2nx

f x

.

Nn theo nh l Bolzano Cauchy: ! 1n

x sao cho 0nf x .

Hay ! 1n

x 2 2

1 1 1 1 1... ... 0

2 1 4 1 1 1x x k x n x

.

Vy phng trnh (1) c duy nht mt nghim ln hn 1 (pcm).

Bi ton 15:

Chng minh rng cc phng trnh sin cosx x v cos sinx x c duy nht

nghim trong 0,2

.

* Phn tch:Bi ton cn chng minh cc phng trnh sin cosx x v

cos sinx x c duy nht nghim trong 0,2

.

Ta cng lm tng t nh nhng bi trc:

+ Chng minh phng trnh cho c nghim trong 0,2

, bng cch s dng

nh l Bolzano Cauchy.

t sin cosf x x x (tng t cos sing x x x ).

+ S dng cng c o hm chng minh hm s f x (tng t g x ) n iu

trn 0,2

.

T y, ta s c c iu phi chng minh.


28/112

26

* Gii:

+ Xt hm s: sin cosf x x x .

Ta c: ' sin .cos cos 1 0f x x x , 0,2

x

.

Ta thy rng hm s f x lin tc v nghch bin trn 0,2

.

Ngoi ra, 0 sin1f v2 2

f

.

Suy ra: 0 . 02

f f

.

Theo nh l Bolzano-Cauchy ta c: ! 0,2

c

sao cho 0f c .

Vy, phng trnh sin cosx x c duy nht nghim trong 0,2

.

+ Tng t, t cos sing x x x .

Hon ton tng t cch trn ta cng chng minh c phng trnh cos sinx x

c duy nht nghim trong 0,2

(pcm).

II.NG DNG NH L LAGRANGE, ROLLE, CAUCHY

1.Phng php chung

ch ra s tn ti nghim ca phng trnh 0k x . (1)

Phng php:

Bc 1:Bin i tng ng v dng 0f x .

V d:

+ Nhn hai v vi tha s khc 0: 1 . 0f x k x g x vi 0g x ;

+ Chuyn v: 0h x k x f x h x k x ;


29/112

27

+ Chia cho mt v (nu c th):

1 0

h xh x k x f x

k x ;

+ Chia cho mtv ri ly cn :

1 0

k

k

h xh x k x f x

k x ;

.....

Bc 2:Dng mt trong cc cch sau y:

Kho st nguyn hm, dng nh l Rolle, nh l Lagrangehoc nh l Cauchy:

t o

x

x

F x f t dt C , v ( )f x lin tc.

nh l Rolle: F b F a th ,c a b ' 0F c f c .

nh l Lagrange: ,c a b sao cho

'F b F a

F c f cb a

.

p dng c kt qu cacc nh l ny vo vic chng minh phng trnh c

nghim th iu quan trng nht l nhn ra c hm F x (thc cht l nguyn hm

ca hm f x ).

*Mt s quy tc o hm tch thng s dng:

. ' . '

. ' . '

' '

' '

. cos sin ' cos . ' sin '

ax ax

ax ax

x x

x x

f x e e a f x f x

f x e e a f x f x

f x e e f x f x

f x e e f x f x

f x ax ax ax f x f x ax f x f x

T , nu thy c ' . 0f c a f c . Ta nn nhn hai v vi axe .

' . 0f c a f c . Ta nn nhn hai v vi axe .

Cc tha s nn nhn vo hai v ca phng trnh l 2 2, , 1,...axe x x


30/112

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2.

Bi tp

Bi ton 1:Chng minh rng nu ' 0f x (hoc ' 0f x ) vi ,x a b th

phng trnh: 1 2 1 2f x f x x x vi mi 1 2, ,x x a b .

* Gii:

Theo bi ra ' 0f x (hoc ' 0f x ) nn f x lin tc v kh vi trn ,a b .

Theo nh l Lagrange:

1 2, ,c x x a b sao cho 2 1 2 1'f c x x f x f x .

Khi , ta c:

1 2 2 1 2 1 2 10 ' 0f x f x f x f x f c x x x x .

Do ' 0f c . Vy ta c iu phi chng minh.

* Nhn xt:Gi s thay hm f x bi hm g x f x x .

Ta c c kt qu nh sau: nu ' 1 0f x (hoc ' 1 0f x ) vi mi

1 2, ,x x a b th phng trnh: 1 1 2 2 1 2f x x f x x x x .

Bi ton 2:Nu ' 1 0f x v ,f x a b vi mi ,x a b th phng trnh

f f x x f x x .

* Gii:Phng trnh: 0f f x x f f x f x f x x . (1)

V ,f x a b (gi thit x f x ) nn theo nh l Lagrange:

,c x f x sao cho: 'f c f x x f f x f x . (2)

Phng trnh (1) v (2) tng ng vi:

' ' 1 0f c f x x f x x f x x f c . (3)

V ' 1 0f c nn phng trnh (3) tng ng vi 0f x x hay l

f x x .

Vy ta c iu phi chng minh.


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* Tng qut:

.......n ln f

f f x x f x x , vi 1,n n .

Bi ton 3:Chng minh rng nu ' 0f x , ,x a b th:

f x f y x y ,

, ,x y a b .

( Ngc li, nu ' 0f x , ,x a b th f x f y x y , , ,x y a b )

* Gii:Gi s x y .

Theo bi ra: ' 0f x , ,x a b nn hm s f x lin tc v kh vi trn ,a b .

Theo nh l Lagrange: ,c x y sao cho:

' f y f x

f cy x

.

Do 0

0

f y f x

y x

suy ra ' 0f c (V l, v bi ton cho ' 0f x ,

,x a b ).

Vy, x y .

Tng t,ta c iu ngc li.

Bi ton 4:Chng minh rng nu 1 2,x x l hai nghim lin tip ca phng trnh

( ) 0f x , vi 1 2x x , 1 2, ( , )x x a b th phng trnh '( ) 0f x c t nht mt nghim

1 2( , ) ( , )ox x x a b .

* Gii:Theo bi ra, 1 2,x x l hai nghim lin tip ca phng trnh ( ) 0f x nn

1 2( ) 0f x f x .

Theo nh l Rolle, ta c: 1 2,ox x x sao cho ' 0of x , hay ta c th ni

phng trnh '( ) 0f x c t nht mt nghim 1 2( , )ox x x .

Mt khc: 1 2x x , 1 2, ( , )x x a b nn 1 2( , ) ( , )ox x x a b .

*Tng qut:Nu phng trnh ( ) 0f x c n nghim phn bit th phng trnh

'( ) 0f x c t nht 1n nghim phn bit. Phng trnh ( ) ( ) 0kf x c t nht n k

nghim phn bit ( 1,2,...)k .


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Bi ton 5:Chng minh rng nu ' 0f x (hoc ' 0f x ), ,x a b .

Phng trnh 0f x c nghim ox th ox l duy nht.

* Gii:Gi s ngc li cn 1x khc ox cng l nghim ca phng trnh

0f x .

Khng mt tnh tng qut, gi s 1ox x

Khi , theo nh l Rolle: 1,oc x x sao cho ' 0f c . iu ny mu thun vi

gi thit ' 0f x , ,x a b .

Vy,ox l nghim duy nht ca phng trnh 0f x .

Bi ton 6:Nu phng trnh ' 0f x c nghim duy nht th phng trnh

0f x c khng qu hai nghim.

* Gii:Gi s phng trnh 0f x c qu hai nghim.

Khng mt tnh tng qut, gi s 1 2 3x x x l nghim phng trnh.

Khi , theo nh l Rolle, 1 1 2,c x x v 2 2 3,c x x sao cho:

1

2

' 0

' 0

f c

f c

Do 1 2 2 3, ,x x x x , suy ra 1 2c c , hay phng trnh ' 0f x c hai

nghim phn bit.

Mu thun vi gi thit l ' 0f x c nghim duy nht.

Vy, ta c iu phi chng minh.

Bi ton 7: Chng minh rng nu '' 0f x (hoc '' 0f x ), ,x a b th

phng trnh 0f x c khng qu hai nghim.

* Gii: V '' 0f x , ,x a b nn theo bi ton 5, suy ra phng trnh

' 0f x nu c nghim ox th ox l duy nht.


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Khi , theo bi ton 6 th phng trnh 0f x c khng qu hai nghim.

Bi ton 8:Chng minh rng phng trnh cos cos2 cos3 0a x b x c x lun c

nghim vi mib cc s thc , ,a b c .

* Phn tch:

Ta thy hm s cos cos 2 cos3g x a x b x c x lin tc v kh vi trn .

chng minh phng trnh c nghim ta c th s dng nh l Rolle. iu ta cn

quan tm by gi l tm c hm f x l nguyn hm ca g x .

Ta c:

.cos .cos2 .cos3 .sin .sin 2 .sin32 3

b cf x g x dx a x b x c x dx a x x x .

Nhvy, hm s f x cn s dng l .sin .sin 2 .sin32 3

b cf x a x x x .

* Gii:

Xt .sin .sin 2 .sin 32 3

b cf x a x x x .

Ta c: ' .cos .cos2 .cos3f x a x b x c x g x , x .

Ta li c: 0 0f f .

By gi, ta xt hm s f x trong on 0, .

Ta thy f x lin tc v kh vi trn 0, .

p dng nh l Rolle: 0 (0; )x sao cho ' 0of x .

Hay 0 (0; )x sao cho ' 0of x .

Suy ra, phng trnh cho lun c nghim vi mi b s thc , ,a b c (pcm).

* Tngqut:

- Phng trnh 1

cos 0n

i

i

a ix

lun c nghim , 1,

ia i n v:


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Xt hm s 1

sinn

i

i

aF x ix

i

kh vi v lin tc trn 0, v:

1

' cos

0 0

n

i

i

F x a ix

F F

Khi 0,ox sao cho:

1

0' cos 0

0

n

o i o

i

F FF x a ix

.

Hay phng trnh 1

cos 0n

i

i

a ix

lun c nghim 0,ox .

- Phng trnh 1

sin 0n

i

i

a ix

lun c nghim , 1,ia i n v:

Xt hm s 1

cosn

i

i

aF x ix

i

kh vi v lin tc trn 0,2 v:

1

' sin

2 0 0

n

i

i

F x a ix

F F

Khi 0,2ox sao cho:

1

2 0' cos 0

2 0

n

o i o

i

F FF x a ix

.

Hayphng trnh 1

sin 0n

i

i

a ix

lun c nghim 0,2ox .

Bi ton 9:Cho s thc dng mv cc s thc a, b, ctha mn:

02 1

a b c

m m m

.

Chng minh rng 2 0ax bx c c nghim thuc 0,1 .

* Phn tch:Phng trnh 2 0ax bx c lin tc v kh vi trn 0,1 .

chng minh phng trnh 2 0ax bx c c nghim thuc 0,1 ta s dng

nh l Rolle.


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+ Nu ly nguyn hm v tri ca phng trnh l 2ax bx c th ta vn cha s

dng c nh l Rolle.

+ iu kin bi ton: 02 1

a b c

m m m

.

Nu nhn hai v ca phng trnh vi mt lng khc 0 l 1mx .

Ta c: 1 2 0mx ax bx c th vic xt nguyn hm ca v tri c th gip ta

s dng c nh l Rolle.

Khi :

2 1

2 1 1 1 . . ..2 1

m m mm m m m a x b x c x

F x ax bx c x dx ax bx cx dxm m m

.

Vy, hm s cn xt l:2 1. . .

( )2 1

m m ma x b x c xF x

m m m

.

* Gii:

- Xt hm s2 1. . .

( )2 1

m m ma x b x c x

F xm m m

lin tc trn 0,1 , kh vi trn 0,1 .

V 1 2' mF x x ax bx c .

Ngoi ra 0 1 0F F .

- p dng nh l Rolle khi : 0,1 sao cho:

1 2

2

' 0 . . 0

. . 0

mF a b c

a b c

Vy phng trnh 2 0ax bx c c nghim trong 0,1 .

* Tng qut:

Cho s thcdng m , s nguyn dng nv cc s thc 0 1, ,..., na a atha mn:

1 0... 01

n na a a

m n m n m

.

Chng minh rng 11 1 0... 0n n

n na x a x a x a

c nghim thuc 0,1 .


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34

Hng dn:Xt hm s 11 0( ) ...1

m n m n mn na a a

f x x x xm n m n m

.

Bi ton 10: Cho 0a b c .

Chng minh rng: sin 9 sin3 25 sin 5 0a x b x c x c t nht 4 nghim thuc

0, .

* Phn tch: chng minh ( )g x c t nht n nghim ta cn chng minh f x c

t nht 1n nghim vi f x l mt nguyn hm ca ( )g x trn ,a b (c th phi p

dng nhiu ln).

- t sin 9 sin 3 25 sin 5g x a x b x c x .

Ta c:

0 .sin 0 9 .sin0 25 .sin 0 0g a b c .

.sin 9 .sin3 25 .sin5 0g a b c .

Nh vy phng trnh 0g x c hai nghim l 0,x x .

Ta ch cn tm trong on 0, , phng trnh 0g x c t nht hai nghim na

th ta c c iu cnchng minh.

- By gi ta i tm nguyn hm ca hm s g x .

Ta c, nu ch xt nguyn hm mt ln ca hm s g x th:

.sin 9 .sin3 25 .sin5 .cos 3 .cos3 5 .cos5g x dx a x b x c x dx a x b x c x h x

Ta thy 0 0h h , nn cha th s dng nh l Rolle.

Ta xt tip nguyn hm ca hm s h x :

.cos 3 .cos3 5 .cos5 .sin .sin 3 .sin 5h x dx a x b x c x dx a x b x c x f x

Vy, hm s cn xt l: ( ) sin sin3 sin5f x a x b x c x .

* Gii:t sin 9 sin 3 25 sin 5g x a x b x c x .


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Ta c:

0 .sin 0 9 .sin0 25 .sin 0 0g a b c .

.sin 9 .sin3 25 .sin5 0g a b c .

Nh vy phng trnh 0g x c hai nghim l 0,x x . (1)

- Xt hm s: ( ) sin sin3 sin 5f x a x b x c x , ta thy f x lin tc v kh vi

trn , nn f x cng lin tc v kh vi trn on 0, .

Ta c: 0 0f f .

3 5 2 2 2 2

.sin .sin .sin 04 4 4 4 2 2 2 2

f a b c a b c a b c

.

3 3 9 15 2

.sin sin sin 04 4 4 4 2

f a b c a b c

.

( Theo gi thit 0a b c ).

- p dng nh l Rolle trn cc on3 3

0, , , , ,4 4 4 4

.

Ta c: 1 2 33 3

0, , , , ,4 4 4 4x x x

Sao cho 1 2 3' ' ' 0f x f x f x .

Vi '( ) .cos 3 .cos3 5 .cos5f x a x b x c x h x .

- Nhn thy, hm s h x cng l hm s lin tc v kh vi trn .

Ngoi ra, 1 2 3' ' ' 0f x f x f x hay 1 2 3 0h x h x h x .

Tip tc p dng nh l Rolle i vi hm s h x trn on 1 2,x x :

4 1 2 5 2 3, , ,x x x x x x sao cho 4 5' ' 0h x h x .

Vi '( ) .sin 9 .sin3 25 .sin5h x a x b x c x g x .

Nh vy, 4 5' ' 0h x h x hay 4 5 0g x g x .


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M 4 1 23

, 0, 0,4

x x x

v 5 2 3, , 0,4x x x

.

Suy ra 4 5,x x l hai nghim ca phng trnh 0g x trong on 0, . (2)

T (1) v (2), ta c iu phi chng minh.

Bi ton 11:Cho hm s f x lin tc v c o hm trn 0, v khng phi

l hm hng.Cho 2 s thc 0 a b .

Chng minh phng trnh:( ) ( )

'( ) ( ) af b bf a

xf x f xb a

c t nht mt nghim

thuc (a;b).

(Olympic sinh vin 1994).

* Phn tch:

Bi ton cn chng minh phng trnh:( ) ( )

'( ) ( ) af b bf a

xf x f xb a

c t nht

mt nghim thuc (a;b).

Bi ton cho hm s f x lin tc v c o hm trn 0, nn ta s s dng

mttrong cc nh l Lagrange, Rolle, Cauchy chng minh s tn ti nghim.

iu quan trng by gi l tm c hm s ph hp p dng c mt trong

cc nh l trn.

Ta c:

( ) ( )

1 1 1 1

f b f a f b f a

af b bf a b a b a

b a

a b b a

Mt khc:

'

2

'( ) ( ) f xxf x f x

x x

T y, ta s xt hai hm s

( ) 1

( ) ; ( )

f x

g x h xx x

ri s dng nh l Cauchy

Vi2 2

'( ) ( ) 1'( ) ; '( )

xf x f xg x h x

x x

Ta s c c iu phi chng minh.

* Gii:


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Xt 2 hm s:( ) 1

( ) ; ( )f x

g x h xx x

th g x , h x lin tc v kh vi trn ,a b

Ta c:2 2

'( ) ( ) 1'( ) ; '( )

xf x f xg x h x

x x

.

Theo nh l Cauchyth

,ox a b sao cho: . ' . 'o oh b h a g x g b g a h x

hay 0 0 02 2

0 0

'( ) ( )1 1 ( ) ( ) 1x f x f x f b f a

b a x b a x

.

Do 2 2

( )( '( ) ( )) ( ) ( )o o o

o o

a b x f x f x af b bf a

bax abx

.

Suy ra 0 0 0 ( ) ( )'( ) ( ) af b bf ax f x f xb a

.

Vy phng trnh:( ) ( )

'( ) ( ) af b bf a

xf x f xb a

c t nht mt nghim thuc

,a b (pcm)

Bi ton 12:

Cho hm s f x kh vi trn 0,1 v tho mn: 0 0; 1 1f f

Chng minh tn ti 2 s phn bit ,a b

thuc 0,1 sao cho ' . ' 1f a f b

* Phn tch:Theo nh l Lagrange, ta ch

cn ch ra im C trn cung OA sao cho

. 1 . 1CF AE KB AB

OF CE OB HB

M 1 1

KB

AB OB KB HBHB

T y, ta chn C l giao im cung OA vi BD hay honh C tha mn

phng trnh: 1f x x .

Hay, hm s cn xt l 1g x f x x .

D

B

A

CEK

FHO


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* Gii:

- Xt hm s 1g x f x x th ( )g x

lin tc v kh vi trn 0,1 .

Ta c: 0 1 0g v 1 1 0g nn

theo nh l Bolzano-Cauchy 0,1c sao cho

0g c .

Do 1 0f c c hay 1f c c .

- p dng nh l Lagrange cho f x trn cc on 0,c v ,1c th:

0,a c sao cho: ( ) (0) '( )0

f c ff a

c

V ,1b c sao cho:(1) ( )

'( )1

f f cf b

c

.

Nn:( ) 1 ( ) (1 )

'( ). '( ) 11 (1 )

f c f c c cf a f b

c c c c

.

Vy tn ti 2 s phn bit ,a b thuc 0,1 sao cho ' . ' 1f a f b (pcm).

Bi ton 13:Cho n l s nguyn dng, , ( 1,2,..., )k k

a b k n .

Chng minh rng phng trnh: 1

sin cos 0n

k k

k

x a kx b kx

c nghim trong

khong , .

* Phn tch: chng minh phng trnh c nghim trong khong , , cng

nh nhng bi trc, ta cn tm nguyn hm ca hm s:

1

sin cosn

k k

k

f x x a kx b kx

Ta c:

2

1 1

sin cos cos sin2

n nk k

k k

k k

a bxx a kx b kx dx kx kx F x

k k

.

D

C

B

A

E

K

FHO


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By gi, hm s ta cn xt l: 2

1

cos sin2

nk k

k

a bxF x kx kx

k k

.

* Gii:

Xt hm

2

1

cos sin ,2

n

k k

k

a bxF x kx kx x

k k

.

R rng F x lin tc trn on ,a b v kh vi trn .

Ta c: 1

' sin cosn

k k

k

F x x a kx b kx

.

Mt khc:

2

1

2

1

12

12

nkk

k

nkk

k

aF

k

aF

k

Ta thy rng : F F .

S dng nh l Rolle:

,c sao cho 1

' 0 sin cos 0n

k k

k

F c c a kc b kc

Hay phng trnh 1

sin cos 0n

k k

k

x a kx b kx

c nghim thuc ,

(pcm).

Bi ton 14: Cho hm s g x lin tc trn 0,1 v kh vi trong 0,1 v tha

mn cc iu kin 0 1 0g g . Chng minh rng tn ti 0,1c sao cho

'g c g c .

* Phn tch:T kt qu cn tm ' ' 0g c g c g c g c . (1)

t 'h x g x g x . Vic xt nguyn hm ca hm s h x ny l kh kh

khn, nhng nu ta p dng cng thc: . ' . 'ax axf x e e a f x f x .


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40

Nhn c hai v ca phng trnh (1) cho xe khi ta c ' . 0x

g x g x e .

Ri p dng cng thc trn th ' . . 'x xg x g x e e g x .

Vy, nguyn hm ca hm s ' . xk x g x g x e l x

f x e g x .

Sau , p dng nh l Rolle cho hm s ny ta s c kt qu cn tm.

* Gii:

Xt hm s: xf x e g x .

Ta thy f x lin tc trn 0,1 v kh vi trong 0,1 .

Ta c: ' ' xf x g x g x e .

Ngoi ra: 00 0 0f e g v 11 1 0f e g .

Nh vy, theo nh l Rolle th:

0,1c sao cho ' 0 ' 0 'cf c g c g c e g c g c .

Tc l 0,1c sao cho 'g c g c .

Vy ta c iu phi chng minh.

Bi ton 15:Cho hm s f x c 'f x ng bin trong ,a b vi:

1 1

,2 2

f a a b f b b a .

Chng minh rng tn ti , , phn bit trong ,a b sao cho:

' . ' . ' 1f f f .

* Phn tch:Theo bi ra, 'f x ng bin trong ,a b nn n lin tc trn .

p dng nh l Lagrange cho hm s f x trn ,a b , ta c:

,a b sao cho:

' f b f a

fb a

.


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41

V

f b f a b a suy

ra: ' 1f .

Bi ton cn chng

minh: , , phn bit

trong ,a b sao cho:

' . ' . ' 1f f f .

M ' 1f nn ta

ch cn chng minh

' . ' 1f f .

Quay tr li vi bi ton 12, ta s s dng phng php dng trc quan xt hm

2

a bg x f x x

.

Ri tip tc lm tng t, ta s c c iu phi chng minh.

* Gii:

Hm s f x c 'f x ng bin trong ,a b nn n lin tc trn , ta cng c

, : ' 'x a b x y f x f y . (1)

Theo nh l Lagrange, th

, : '( ) 1f b f a

a b fb a

. (2)

S dng phng php trc quan tng t bi ton 12. Ta xt hm

2

a bg x f x x

.

Hm s g x lin tc trn ,a b v 2

. 0g a g b a b .

Suy ra rng: ,ox a b sao cho 0og x 02o oa b

f x x

.

O

A

D C

B

y

x

a b

1

2f a a b

1

2f b b a

K


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42

Hay 2o o

a bf x x

.

Tip tc p dng nh l Lagrange, , , ,o oa x x b sao cho:

' . ' . 1o o

o o

f x f a f b f x

f f x a b x

. (3)

T (1),(2), (3) ta suy ra , phn bit trong ,a b v ' . ' . ' 1f f f .

( V nh ta c , gi s ' ' ' 1f f f .

Do 'f x ng bin nn , mu thun.

Tng t cho trng hp .

T y ta suy ra: , , phn bit trong ,a b )

Ta c c iu phi chng minh.

Bi ton 16: Cho hm s f x xc nh v lin tc trn on ,a b , ( )a b v

tha mn iu kin 0b

a

f x dx .

Chng minh rng: ,c a b sao cho 2005

c

af c f x dx .

* Phn tch:

Bi ton cn chng minh ,c a b sao cho 2005c

a

f c f x dx .

Tc l chng minh phng trnh 2005 0t

a

f t f x dx c nghim thuc ,a b .

Theo bi ra, ta c hm s f x lin tc v c o hm trn ,a b .

Nh vy, chng minh s tn ti nghim ca phng trnh trn, ta c th s dng

1 trong cc nh l Lagrange, Rolle, Cauchy. iu quan tm by gi l tm c mt

hm ph c th s dng c 1 trong cc nh l .


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43

+ Nu t h t 2005t

a

f t f x dx , ta nhn thy cha th ly nguyn hm ca

hm s ny c.

+ Nhng nu ta nhn vo phng trnh 2005 0t

a

f t f x dx

mt lng khc 0

l 2005te ta c 2005 2005 0t

t

a

e f t f x dx

.

t: h t 2005 2005t

t

a

e f t f x dx

'

2005t

t

a

e f x dx

.

Ta thy ngay: h t dt 2005

t

t

a

e f x dx g t .

Vy, hm s cn xt l: 2005t

t

a

g t e f x dx .

* Gii:

Xt hm s: 2005t

t

a

g t e f x dx .

Khi : 0F a F b v 2005 2005' 2005

t

t t

a

g t e f x dx e f t .

Theo nh l Rolle, tn ti ,c a b sao cho ' 0g c , ngha l:

2005 20052005 0c

c c

a

e f x dx e f c .

Hay t y suy ra iu phi chng minh: 2005

c

af c f x dx

.

Bi ton 17: Gi s f x l hm kh vi trn ,a b , tha mn iu kin

0; 0, ,f a f b f x x a b . Chng minh rng tn ti dy nx vi

,nx a b sao cho:

'

lim 20131

n

nnn

f x

e f x

.


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* Phn tch:

Bi ton cn chng minh ,nx a b sao cho:

'

lim 20131

n

nnn

f x

e f x

.

T kt qu

'

lim 20131n

nnn

f x

e f x th ta cn chng minh c

'

20131

n

n

n

f x

e f x

' 2013 1 .nn nf x e f x .

Ta c: ' 2013 1 . ' 2013 1 . 0n nf x e f x f x e f x . (1)

Vic xt nguyn hm ca hm s h x ny l kh kh khn, nhng nu ta p dng

cng thc: . ' . 'ax axf x e e a f x f x .

Nhn c hai v ca phng trnh (1) cho 1 2013n e x

e

khi ta c:

1 2013' 2013 1 . . 0n e xnf x e f x e

.

Ri p dng cng thc trn th

1 2013 1 2013

' 2013 1 . . .

n ne x e xn

f x e f x e e f x g x

.

Nguyn hm ca hm s 1 2013' 2013 1 . . n e xnh x f x e f x e

l

1 2013

.n e x

g x e f x

.

Vy, hm s cn xt l

1 2013

.n e x

g x e f x

.

Ri p dng nh l Rolle cho hm s ny ta s c kt qu cn tm.

* Gii: n

t: 1 2013

.n e x

g x e f x

.

D thy rng g x lin tc v kh vi trn ,a b , ngoi ra:


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45

1 2013

1 2013

. 0

0

0

n

n

e a

e b

g a e f a

g b e f b

g a g b

Nh vy theo nh l Rolle th ,nx a b

' 0n

g x .

Ta c: 1 2013

' ' 1 2013.n e x ng x e f x e f x

.

Nn

' 0 ' 1 2013. 0

'2013 2013

1 .

n

n n n

n

n

n

g x f x e f x

f xn

e f x

Vy

'

lim 20131

n

nnn

f x

e f x

.

Bi ton 18: Cho f x lin tc trn 0,a , kh vi trn 0,a sao cho 0f a .

Chng minh rng tn ti 0,c a 1

' c

f c f cc

.

* Phn tch:

Bi ton cn chng minh rng tn ti 0,c a :

1

' . ' 1 0c

f c f c c f c f c cc

.

Tc l chng minh phng trnh . ' 1 0x f x f x x c nghim 0,c a .

Theo bi ra f x lin tc trn 0,a , kh vi trn 0,a . Nn chng minh

phng trnh trn c nghim, ta c th s dng mt trong cc nh l Lagrange, Rolle,

Cauchy. s dng c cc nh l ny iu u tin l phi tm c nguyn hm

ca hm s . ' 1h x x f x f x x .

Ta thy rng khng th ly nguyn hm trc tip c, v vynhn c hai v ca

hm s vi mtlng khc 0 l xe .


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Ta c: . . . ' 1x xe h x e x f x f x x .

Ta c: . . . ' 1x xe h x dx e x f x f x x dx .

' . .x xe f x f x x dx e f x xdx . (1)

* Tnh . 'xe f x x f x dx .

t ' . .

x xu e du e dx

dv f x f x x dx v x f x

Suy ra: . ' . . . .x x xe f x x f x dx e x f x x f x e dx . (2)

Th (2) vo (1): .xe h x dx . . x

x f x e g x .

Nh vy, hm s cn xt l . . xg x x f x e .

* Gii: Xt hm s . . xg x x f x e lin tc trn 0,a .

Ta c: 0 0g g a ; g x kh vi trn 0,a .

Theo nh l Rolle, 0,c a sao cho ' 0g c .

Ta li c: ' . . 'x

g x e x f x f x x f x

.

Nn ' 0 . . ' 0g c c f c f c c f c 1

' .c

f c f cc

.

Hay 0,c a sao cho 1

' c

f c f cc

(pcm).


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Bi ton 19:Cho f x lin tc trn on 0,2

, kh vi trn khong 0,2

sao

cho 0 02

f f

v 22 ' 0, 0,

2f x f x x

.Chng minh rng tn ti

0, 2c :

a/

'tan

'

f c f cc

f c f c

. b/

22 2

. '1cos

2 '

f c f cc

f c f c

.

* Phn tch:

a/ T kt qu cn tm:

'tan

'

f c f cc

f c f c

ta bin i a v dng

'sinsin . ' cos . '

cos '

cos ' sin ' 0

cos ' sin ' 0

f c f ccc f c f c c f c f c

c f c f c

c f c f c c f c f c

c f c f c c f c f c

Ta xt hm s cos ' sin 'h x x f x f x x f x f x .

p dng cng thc:

. cos sin ' cos . ' . sin ' .f x ax ax ax f x a f x ax f x a f x .

tm ra nguyn hm ca hm s h x l . cos sing x f x x x .

b/ Bin i kt qu c cu a/ tm ra kt qu.

*Gii:

a/ Xt hm s: . cos sing x f x x x .

Hm s g x lin tc trn 0, 2

, kh vi trn 0,

2

, 0 02

g g

.

Theo nh l Rolle, 0,2

c

' 0g c .

Ta c: ' ' cos sin cos sing x f x x x f x x x .


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Vy ' 0 cos ' sin 'g c c f c f c c f c f c . (*)

Nu ' 0f c f c th t (*) suy ra ' 0f c f c .

Vy ' 0f c f c , iu ny mu thun.

Nn ' 0f c f c , ta chia c hai v ca (*) cho ' cos 0f c f c c .

Ta c:

cos ' sin '

cos ' cos '

'tan

'

c f c f c c f c f c

c f c f c c f c f c

f c f cc

f c f c

Ta c iu phi chng minh.

b/ T kt qu ca cu a:

'tan

'

f c f cc

f c f c

.

Ta c:

22 2

22 2 2

2 22

2 2

2 2

2 2

' ' 2 . 'tan

' 2 . ''

' 2 . '1 tan 1' 2 . '

2 '

' 2. . '

f c f c f c f c f c f cc

f c f c f c f cf c f c

f c f c f c f ccf c f c f c f c

f c f c

f c f c f c f c

V 2 22 2

1 11 tan cos

cos 1 tanc c

c c

2 22

2 2

' 2. . 'cos

2 '

f c f c f c f cc

f c f c

22 2

. '1cos

2 '

f c f cc

f c f c

Vy

2

2 2

. '1cos

2 '

f c f cc

f c f c

.


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Bi ton 20:Cho , , 0a b c tha mn 07 5 3

a b c .

Chng minh rng th hm s 4 2. .f x a x b x c lun ct trc Ox ti t nht

mt im c honh nm trn khong 0,1 .

* Phn tch: chng minh th hm s 4 2. .f x a x b x c lun ct trc Ox

ti t nht mt im c honh nm trn khong 0,1 .

Ta phi chng minh rng phng trnh 4 2. . 0a x b x c c nghim trn 0,1 .

Vic xt nguyn hm ca v tri 4 2. .a x b x c khng gip ch g cho ta v cha s

dng c nh l Rolle.

iu kin cho 07 5 3a b c .

Nu nhn hai v vi lng khc 0 l 2x , phng trnh tng ng vi

6 4 2. . . 0a x b x c x th vic kho st nguyn hm ca v tri rt c ch.

Ta c: 6 4 2 7 5 3. . . . . .7 5 3

a b ca x b x c x dx x x x g x .

*Gii:Xt hm

7 5 3. . .

7 5 3

a x b x c x

g x .

Ta thy rng g x kh vi lin tc trn on 0,1 v 0 1 0g g .

Theo nh l Rolle: 0,1ox 4 2' 0 . . 0

o o og x a x b x c .

Vy th hm s 4 2. .f x a x b x c lun ct trc Ox ti t nht mt im c

honh nm trn khong 0,1 .

*Tng qut:Cho 1n s thc 1, ,....,o nc c c tha mn:

1

1 1.... 0

2 1o nc c c

n

Chng minh rng phng trnh 1 .... 0n

o nc c x c x c t nht mt nghim trong

khong 0,1 .


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Hng dn:Xt hm s 2 1

1 ...2 1

n

o n

x xg x c x c c

n

. (S dng nh l Rolle).

Bi ton 21: Cho hm s f x xc nh v lin tc trn ,a b sao cho

' , "f x f x lin tc trn ,a b , 0f a f b .

Chng minh rng , , ,x a b z x a b :

. ''2

x a x bf x f z x

. (*)

* Phn tch:R rng khng nh ng vi ,x a x b nn ta ch vic chng minh

cho ,x a b .

Tt nhin

(*) '' 02

x a x bf x f z x

. (1)

Nu ta nhn v tri ca (1) nh l gi tr ca hm

. ''2

t a t bf t f z t

ti

im t x th vic p dng nh l Rolle khng thnh cng bi v vn cn z t trong

biu thc ''f .

Ta mong mun p dng nh l Rolle (c thl nhiu ln) cho hm s g x cho

ph hp.

By gi ta nhn (*) di dng

2''

f xf z x

x a x b

hay

2'' 0

t z x

f xf t

x a x b

.

T , hm g t ph hp ( ly nguyn hm hai ln) l:

f xg t f t t a t b

x a x b

.

* Gii:


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Xt hm s:

f xg t f t t a t b

x a x b

.

Ta c 0g a g x g b , nn theo nh l Rolle, 'g t trit tiu ti t nht hai

im l 1 ,c a x v 2 ,c x b .

Ta li c: theo nh l Rolle, ''g t b trit tiu t nht ti mt im

1 2, ,z z x c c a b . Tc l '' 0g z x .

V

'' '' 2

f xg t f t

x a x b

.

Nn

'' 0 '' 2. 0f x

g z x f z xx a x b

.

'' 2. . ''

2

f x x a x bf z x f x f z x

x a x b

.

Vy, , , ,x a b z x a b sao cho

. ''2

x a x bf x f z x

(pcm).

Bi ton 22:Cho f x l mt hm lin tc trn 0,1 , kh vi trn 0,1 , 1 0f .

Chng minh rng tn ti 0,1c sao cho: 1 . ' 02013f c c f c .

* Phn tch:

Ta cn chng minh 0,1c sao cho 1

. ' 02013

f c c f c

2013. . ' 0f c c f c .

Tc l, cn chng minh phng trnh 2013. . ' 0f x x f x

c nghim

0,1c . V theo bi ra f x l mt hm lin tc trn 0,1 , kh vi trn 0,1 nn ta c

th s dng mttrong cc nh l Lagrange, Rolle, Cauchy chng minh.

s dng c mttrong cc nh l ny, ta phi tm c hm s ph hu ch.


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Ta cha th ly nguyn hm trc tip ca hm s 2013. . 'f x x f x , nhng nu

nhn vo hai v ca phng trnh 2013. . ' 0f x x f x vi mt lng khc 0 l

2012x , ta c:

2012 2012 20132013. . ' 0 2013. . . ' 0x f x x f x x f x x f x .

t 2012 20132013. . 'h x x f x x f x .

Ly nguyn hm ca hm s h x ta c: h x dx 2013.x f x g x .

Vy, hm s cn xt l 2013.g x x f x .

* Gii:

Xt hm s 2013

.g x x f x .

Ta thy, g x l hm lin tc trn 0,1 , kh vi trn 0,1 .

Ta c: 20130 0 . 0 0g f v 20131 1 . 1 0g f .

Suy ra: 0 1 0g g .

p dng nh l Rolle, 0,1c sao cho ' 0g c .

Ta li c: 2012 2013' 2013. . ' .g x x f x f x x .

Nn : 2012 2013' 0 2013. . ' . 0g c c f c f c c .

1

2013. ' . 0 . . ' 02013

f c f c c f c c f c .

Vy, tn ti 0,1c sao cho 1

. ' 02013

f c c f c (pcm).

Bi ton 23:

Cho , 1 , f x kh vi trn 0,1 , 0 0f v 0, 0,1f x x . Chng

minh rng tn ti 0,1c sao cho:

' ' 1. .

1

f c f c

f c f c

.


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* Phn tch:

Bi ton cn chng minh tn ti 0,1c sao cho:

' ' 1. .

1

' ' 1. . 0

1

' 1 ' 1 0

f c f c

f c f c

f c f c

f c f c

f c f c f c f c

Tc l chng minh phng trnh ' 1 ' 1 0f x f x f x f x c nghim

trong 0,1 .

Theo bi ra: f x kh vi trn 0,1 nn ta c th ngh ti vic s dng mt trong

cc nh l Lagrange, Rolle, Cauchy chng minh phng trnh trn c nghim.

By gi, ta phi tm c hm ph tha mn yu cu ca mt trong cc nh l ny,

ng thi khi ly o hm s cho ta phng trnh cn chngminh.

Ta thy cha th ly ngay nguyn hm ca v tri ca phng trnh trn l

' 1 ' 1f x f x f x f x .

Nhng nu ta nhn c hai v ca phng trnh trn vi mt lng khc 0 l

1 1. 1f x f x ta c:

1 1

. ' . 1 . ' 1 . 1 . 0f x f x f x f x f x f x

.

Lc ny, ly nguyn hm ca hm s :

1 1

. ' . 1 . ' 1 . 1 .h x f x f x f x f x f x f x

.

Ta c: h x dx

. 1f x f x g x

.

Nh vy, hm s cn xt l: . 1g x f x f x

.

* Gii:Xt hm s: . 1g x f x f x

.

Hm s g x lin tc v kh vi trn 0,1 .


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Ta c:

+ Ti 0x : 0 0 . 1 0g f f

.

+ Ti 1x : 1 1 . 0 0g f f

.

(Theo bi ra: 0 0f ).

Suy ra: 0 1g g .

p dng nh l Rolle: 0,1c sao cho ' 0g c .

Vi 'g x 1 1

. ' . 1 . ' 1 . 1 .f x f x f x f x f x f x

1 1. 1 ' 1 ' 1f x f x f x f x f x f x

.

Hay, 0,1c :

1 1. 1 ' 1 ' 1 0

' 1 ' 1 0

' ' 1. .

1

f c f c f c f c f c f c

f c f c f c f c

f c f c

f c f c

Vy: 0,1c sao cho

' ' 1. .1

f c f cf c f c

(pcm).

Bi ton 24:Cho 1 2 2013, ,...,c c c l cc s thc thamn:

1 3 5 7 2011 20133 5 7 ... 2011 2013 0c c c c c c .

Chng minh rng: 2 21 2 2013.cos 2 .cos 2 ... 2013 .cos2013 0c x c x c x c t nht 4

nghim trn , .

* Phn tch:Bi ton cn chng minh phng trnh

2 21 2 2013.cos 2 .cos2 ... 2013 .cos2013 0c x c x c x c t nht 4 nghim trn

,

+ Nu ta chia khong ra ri s dng nh l Bolzano-Cauchy chng minh tn ti

nghim l kh kh khn, v ta kh xc nh c du ca cc gi tr.


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+ Tuy nhin, nhn vo iu kin ca bi ton:

1 3 5 7 2011 20133 5 7 ... 2011 2013 0c c c c c c .

Ta s ngh ti vic ly nguyn hm ca hm s:

2 2

1 2 2013.cos 2 .cos2 ... 2013 .cos2013f x c x c x c x

.

Ri s dng mt trong cc nh l Lagrange, Rolle, Cauchy c c iu phi

chng minh.

Ta c: f x dx 1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013c x c x c x g x .

Nh vy, hm s cn xt l 1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013g x c x c x c x .

* Gii: Xt hm s 1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013g x c x c x c x .

Hm s g x lin tc v c o hm trn , .

Ta c:

+Ti 0x : 1 2 20130 .sin 0 2 .sin 2.0 ... 2013 .sin 2013.0 0g c c c .

+Ti2

x :

1 2 2013.sin 2 .sin 2. ... 2013 .sin 2013.2 2 2 2g c c c

1 3 5 7 2011 20133 5 7 ... 2011 2013 0c c c c c c .

+ Ti x : 1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013 0g c c c .

+ Ti2

x

:

1 2 2013.sin 2 .sin 2. ... 2013 .sin 2013.2 2 2 2g c c c

1 2 2013.sin 2 .sin 2. ... 2013 .sin 2013.2 2 2c c c

1 3 5 7 2011 20133 5 7 ... 2011 2013 0c c c c c c .

+ Ti x :

1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013g c c c


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1 2 2013.sin 2 .sin 2 ... 2013 .sin 2013 0c c c .

Suy ra: 02 2

g g g g g

.

p dng nh l Rolle: 1 2 3 4, , ,0 , 0, , ,2 2 2 2c c c c

.

Sao cho 1 2 3 4' ' ' ' 0g c g c g c g c .

Vi 2 21 2 2013' .cos 2 .cos2 ... 2013 .cos 2013g x c x c x c x f x .

Xt hm s 2 21 2 2013.cos 2 .cos2 ... 2013 .cos2013f x c x c x c x .

Hm s f x lin tc v kh vi trn , .

Ta c:

1 2 3 4' ' ' ' 0g c g c g c g c hay 1 2 3 4 0f c f c f c f c .

Nh vy, phng trnh 2 21 2 2013.cos 2 .cos2 ... 2013 .cos2013 0c x c x c x c t nht

4 nghim trn , (pcm).

Bi ton 25: Cho 1 2, ,..., na a a tha mn 0

1 02 1

nk k

k

a

k

.Chng minh rng

phng trnh: 0

.cos 2 1 0n

k

k

a k x

c nghim trong 0, 2

.

* Phn tch: Bi ton cn chng minh phng trnh: 0

.cos 2 1 0n

k

k

a k x

c

nghim trong 0,2

.

Vi bi ton ny, vic s dng nh l Bolzano-Cauchy l kh kh khn v vic khxc nh du ca cc gi tr.


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Tuy nhin, vi iu kin bi ton cho 0

1 02 1

nk k

k

a

k

, ta c th ngh ti vic

tm mt hm s l nguyn hm ca hm s 0

.cos 2 1n

k

k

f x a k x

s dng c

mttrong cc nh l Lagrange, Rolle, Cauchy chng minh.

Ta c: f x dx

0

sin 2 1

2 1

nk

k

a k xg x

k

.

Nh vy, hm s cn xt l:

0

sin 2 1

2 1

nk

k

a k xg x

k

.

* Gii: Xt hm

0

sin 2 1

2 1

nk

k

a k xg x

k

.

Hm s g x lin tc v c o hm trn 0,2

.

Ta c:

+ Ti 0x :

0 0

sin 2 1 .0 .sin00 0

2 1 2 1

n nk k

k k

a k ag

k k

.

+Ti2

x :

0 0

sin 2 1 .

2 1 02 2 1 2 1

kn n

k k

k k

a k

agk k

.

( Theo iu kin cho).

p dng nh lRolle: 0,2

c

sao cho ' 0g x .

Vi 0

' .cos 2 1n

k

k

g x a k x f x

.

Vy, 0,2

c

sao cho 0

.cos 2 1 0n

k

k

a k x

.

Hay phng trnh 0

.cos 2 1 0n

k

k

a k x

c nghim trong 0, 2

(pcm).


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Bi ton 26: Cho f x l hm kh vi trn 0,1 , 0 0, 1 1f f .

Chng minh rng 1 2 1 2, 0, , 0,1K K x x sao cho 1 2x x v

1 2

1 21 2' '

K KK K

f x f x .

* Phn tch:Bi ton cn chng minh 1 2 1 2, 0, , 0,1K K x x sao cho 1 2x x

v

1 21 2

1 2' '

K KK K

f x f x .

Bi ton cho f x l hm kh vi trn 0,1 , nn ta c th ngh ti vic s dng mt

trong cc nh l Lagrange, Rolle, Cauchy.

V bi ton cn chng minh trn 0,1 c hai nghim

1 2

,x x tha mn iu kin

bi ton. Nn s dng c mt trong cc nh l trn th ta phi tm 0,1c sao

cho 1 20, , ,1x c x c .

Gi s c im c cn tm, iu ta quan tm by gi l tm c f c .

p dng nh l Lagange cho hm s f x trn 2 on 0, , ,1c c , ta c:

+

1 1 1

0

0, : ' '

f c f f c f c

x c f x cc c f x

. (1)

+

2 2 2

1 1 1,1 : ' 1

1 1 '

f f c f c f cx c f x c

c c f x

. (2)

Cng (1) v (2) v theo v, ta c:

1 2

11

' '

f c f c

f x f x

. (*)

- T kt qu cn chng minh:

1 2 1 2

1 21 2 1 2 1 1 2 2

1' ' ' '

K K K K K K

f x f x K K f x K K f x

.

i chiu vi (*) ta thy: 11 2

Kf c

K K

.


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T cch phn tch ny, c c kt qu cn chng minh, u tin ta phi ch ra

im 0,1c sao cho 11 2

Kf c

K K

.

T y, ta s bt u xt hm s 1

1 2

Kg x f x

K K

.

* Gii:Xt hm 11 2

Kg x f x

K K

.

Ta c: 1 21 2 1 2

0 0, 1 0K K

g gK K K K

.

V 0 . 1 0g g , nn theo nh l Bolzano-Cauchy 0,1c sao cho:

11 2

0 K

g c f cK K .

- p dng nh l Lagrange cho hm s f x trn 0,c ta c:

1 10, : 0 ' .x c f c f f x c .

Do ,

1 11

1 2 1 2 1

' .. '

K Kf x c c

K K K K f x

. (1)

- p dng nh l Lagrange cho hm s f x trn ,1c ta c:

2 2,1 : 1 ' . 1x c f f c f x c .

Do ,

1 22

1 2 1 2 2

1 ' . 1 1. '

K Kf x c c

K K K K f x

. (2)

Ly (1)+(2), ta c:

1 2

1 1 2 2 1 2

1' . '

K K

f x K K f x K K

1 2

1 21 2' '

K KK K

f x f x .

Vy, 1 2 1 2, 0, , 0,1K K x x sao cho 1 2x x v 1 2

1 21 2' '

K KK K

f x f x

(pcm).


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Bi ton 27: Cho f x l hm kh vi lin tc n cp hai trn ,a b sao cho

' ' 0f a f b f a f b .

Chng minh rng ,c a b sao cho ''f c f c .

* Phn tch:Theo gi thit hm s f x l hm kh vi lin tc n cp hai trn

,a b . V vy, chng minh s tn ti nghim ta c th s dng mttrong cc nh l

Lagrange, Rolle, Cauchy. s dng c cc nh l ny, u tin phi nhn ra c

nguyn hm ca hm cn chng minh.

T iu phi chng minh ,c a b sao cho ''f c f c . Nn ta s chng minh

phng trnh '' 0f x f x c nghim ,c a b .

t ''h x f x f x .

Ta khng th ly nguyn hm trc tip ca hm s h x c, v vy ta nhn vo

hm s h x mt lng khc 0 l xe , ta c:

. . '' '' ' '

'' ' ' . ' ' . '

x x x

x x x x

e h x e f x f x e f x f x f x f x

e f x f x e f x f x e f x e f x

Ly nguyn hm:

. . ' . . 'x x x xe h x dx e f x e f x e f x f x g x .

Vy, hm s cn xt l 'xg x e f x f x .

* Gii:Xt hm 'xg x e f x f x .

Hm s g x lin tc v kh vi trn ,a b .

Vi ' ''g x f x f x .

Ta c: ' 0ag a e f a f a v ' 0b

g b e f b f b .

( Theo gi thit ' ' 0f a f b f a f b ).

p dng nh l Rolle: ,c a b sao cho:


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' 0g c '' 0f c f c ''f c f c .

Hay ,c a b sao cho ''f c f c (pcm).

Bi ton 28:Cho f x l hm kh vi lin tc n cp hai trn ,a b v trn on

ny f x c khng t hn ba khng im khc nhau.

Chng minh rng ,c a b sao cho: '' 2. 'f c f c f c .

* Phn tch:Theo gi thit hm s f x l hm kh vi lin tc n cp hai trn

,a b . V vy, chng minh s tn ti nghim ta c th s dng 1 trong cc nh l

Lagrange, Rolle, Cauchy. s dng c cc nh l ny, u tin phi nhn ra c

nguyn hm ca hm cn chng minh.

Theo kt qu cn chng minh l ,c a b sao cho: '' 2. 'f c f c f c .

Vy ta phi chng minh phng trnh '' 2. ' 0f x f x f x c nghim

,c a b .

t '' 2. 'k x f x f x f x .

Nu nh vy th cha th tm c nguyn hm ca hm s k x .

Ta c: '' 2. ' ' '' 'f x f x f x f x f x f x f x . (1)

Nhn vo c hai v ca (1) vi mt lng khc 0 l xe :

. ' . '' 'x xe f x f x e f x f x .

p dng cng thc: . ' ' .ax axe f x e f x a f x .

Suy ra: . ' . ', '' ' . ' 'x x x xe f x f x e f x e f x f x e f x .

Vy . ' . '' ' 'x x xe f x f x e f x f x dx e f x f x h x .

- Ta li c: . ' . 'x xh x e f x f x e f x .

Suy ra: .xh x dx e f x g x .


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Vy hm s cn xt l .xg x e f x .

* Gii:Xt hm s .xg x e f x .

Hm s g x lin tc v kh vi trn ,a b (V f x , xe l hm kh vi lin tc

trn ,a b ).

Theo bi ra f x c khng t hn ba khng im khc nhau trn ,a b nn theo h

qu 2 ca nh l Rolle ta tm c 1 2, ,c c a b , 1 2c c sao cho

1

2

1 1 1 1 1

2 2 2 2 2

' ' 0 '

' ' 0 '

c

c

g c e f c f c f c f c

g c e f c f c f c f c

Tip tc t: ' 'xh x e f x f x g x .

Hm s h x lin tc v kh vi trn ,a b .

Ta c 1 2, ,c c a b , 1 2c c sao cho 1 2' 0, ' 0g c g c hay

1 20, 0h c h c .

p dng nh l Rolle: 1 2, ,c c c a b sao cho ' 0h c .

Vi ' '' 2. 'h x f x f x f x .

Suy ra: ' 0h c '' 2. ' 0 '' 2. 'f c f c f c f c f c f c .

Vy, 1 2, ,c c c a b sao cho '' 2. 'f c f c f c (pcm).

Bi ton 29: Cho f x l mt hm lin tc trn ,a h a h , kh vi trn

, , 0a h a h h . Chng minh rng 0,1 sao cho:

' . ' .f a h f a h h f a h f a h

.

* Phn tch:Bi ton cn chng minh 0,1 sao cho:

' . ' .f a h f a h h f a h f a h .

Theo bi ra, f x lin tc trn ,a h a h , kh vi trn , , 0a h a h h .


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Nn c iu cn chng minh, ta s s dng mt trong cc nh l Lagrange,

Rolle, Cauchy.

Tuy nhin, nu ta xt hm s thng thng l f x trong on ,a h a h th ta

vn cha c iu cn chng minh. V vy, phi xt mthm khc trn mton khc.

thy v tri c cha h m v phi l .h nn ta s xem h lm n v xt hm

s: , 0,g x f a x f a x x h .

* Gii:t , 0,g x f a x f a x x h .

Ta c g x lin tc trn 0,h v kh vi trn 0,h .

Vi: ' ' 'g x f a x f a x .

Theo nh l Lagrange: 0,1 sao cho:

0 ' . . ' . ' . .g h g g h h f a h f a h f a h f a h h .

Bi ton 30: Cho f x l mt hm lin tc v kh vi n cp hai trn ,a b .

Chng minh rng ,c a b sao cho:

2

. ''2 2 8

f a f b b aa bf f c

.

* Phn tch:

Nhn vo biu thc

2

. ''2 2 8

f a f b b aa bf f c

ta thy ''f c

c xc nh thng qua mt i lng c nh. Nn ta s t i lng c nh l

A , ta s tm c sao cho ''f c A .

Biu thc c vit li l 2

.2 2 8

f a f b b aa bf A


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Theo bi ra, f x l mt hm lin tc v kh vi n cp hai trn ,a b . V vy,

chng minh ,c a b sao cho:

2

. ''2 2 8

f a f b b aa bf f c

, ta c th

ngh ti vic dng mt trong cc nh l Lagrange, Rolle, Cauchy.

Nhn thy, trong biu thc

2

.2 2 8

f a f b b aa bf A

vai tr ca a v

b l nh nhau. Ta chn b lm n, tac hm s:

2

.2 2 8

f a f x x aa xF x f A

Vy, ta s bt u xt hm s:

2

.

2 2 8


* Gii:Gi Al hng s sao cho

2

.2 2 8

f a f b b aa bf A

.

t:

2

.2 2 8


.

Vi 1 1 1

' ' ' .2 2 2 4

a xF x f x f x a A

Ta c: 0F a F b do , ,a b sao cho:

1

' 0 ' ' . 02 2 4

a aF f f A

. (*)

Li p dng nh l Lagrange cho hm 'f x trn ,2

a

ta tm c ,c a b

sao cho ' ' '' .2 2a af f f c

.

Thay vo (*) ta c ''f c A .

Nh vy, tn ti ,c a b sao cho

2

. ''2 2 8

f a f b b aa bf f c

.


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Bi ton 31:Cho , ,a b c tha mn2

3 5 2

c a b

n

Chng minh rng phng trnh .sin .cos .sin 0n na x b x c x c c nghim trong

0,

2

.

* Phn tch:

Bi ton cn chng minh phng trnh .sin .cos .sin 0n na x b x c x c c

nghim trong 0,2

.

+ Nu t .sin .cos .sinn nf x a x b x c x c , f x lin tc trn 0,2

.

Ta c:

0 .sin 0 .cos 0 .sin 0n nf a b c c b c .

.sin .cos .sin2 2 2 2

n nf a b c c a c

.

Ta cha th xc nh c du ca tch 0 .2

f f

, nn khng th s dng nh

l Bolzano-Cauchy.

+ Tuy nhin, nu nhn vo iu kin bi ton:

2 6

3 5 2 5 2

c a b a bc

n n

.

Ta s ngh ti s dng mt trong cc nh l Lagrange, Rolle, Cauchy chng

minh phng trnh cho c nghim, bng cch a phng trnh cho v dng:

2sin cos .sin .cos .sin 0n nx x a x b x c x c .

t: 2sin cos .sin .cos .sinn nh x x x a x b x c x c .

Ri ly nguyn hm ca hm s h x , ta c:


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h x dx 2 2 3

22 .sin 2 .cos 2 .sin .cos2 2 3

n na x b x c x

c x g xn n

.

Vy, hm s cn xt l 2 2 3

22 .sin 2 .cos 2 .sin .cos2 2 3

n na x b x c xg x c x

n n

.

* Gii:

Ta c:2 6

3 5 2 5 2

c a b a bc

n n

.

Xt hm 2 2 3

22 .sin 2 .cos 2 .sin .cos2 2 3

n na x b x c xg x c x

n n

.

Ta thy rng g x xc nh, lin tc v c o hm trn 0,2

.

V

1 1 2' 2 .sin .cos 2 .cos .sin 2 .sin .cos 2 .cos .sin

' 2sin .cos .sin .cos .sin

n n

n n

g x a x x b x x c x x c x x

g x x x a x b x c x c

Ta li c:

6 6 40

2 10 2 5 2

a bb a bg

n n n

.

6 42 2 2 6 4.2 2 3 5 2 2 5 2 5 2

a b a ba a a bgn n n n n

.

Suy ra: 02

g g

.

Theo nh l Rolle th 0, : ' 02o o

x g x

.

Hay 0, : ' 2sin .cos .sin .cos .sin 02n n

o ox g x x x a x b x c x c

.

Suy ra: 0,2o

x

sao cho .sin .cos .sin 0n na x b x c x c (pcm).


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Bi ton 32:Gi s f x l hm kh vi n cp 1n trn . Chng minh rng

vi mi cp s thc , ;a b a b sao cho

' ...ln

' ...

n

n

f b f b f b

Định Lý Lagrange, Rolle, Cauchy Và Ứng Dụng

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Transcript of Định Lý Lagrange, Rolle, Cauchy Và Ứng Dụng