Bài tập giải tích về liên tục hàm số

8/2/2019 Bi tp gii tch v lin tc hm s

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BI TP V HM S VI BA VN LIN TC, KH VI, KH TCH

Bi 1. Tm tt c cc hm s ( )u x tha mn ( ) ( )

1

2

0

u x x u t dt = + .

Gii

V ( )

1

2

0

u t dt l mt hng s nn ( )u x x C = + (C l hng s).

Do ( )

112 22

0 0

1 1

2 8 2 4

t Ct C dt C Ct C C C

+ = + = + = =

.

Vy ( ) 14

u x x= + l hm s cn tm.

Bi 2. Cho hm s :f tha mn iu kin: ( ) ( )19 19 f x f x+ + v

( ) ( )94 94 f x f x+ + vi mi x. Chng minh rng: ( ) ( )1 1 f x f x+ = + vi

mi x .GiiLy mt s thc x bt k. p dng iu kin ban cho vi 19x v

94x ta thu c:( ) ( )19 19 f x f x v ( ) ( )94 94 f x f x .

By gita d dng chng minh bng quy np vi mi n ( ) ( )19 19 f x n f x n+ + , ( ) ( )94 94 f x n f x n+ +

( ) ( )19 19 f x n f x n , ( ) ( )94 94 f x n f x n .

Ta c:( ) ( ) ( ) ( ) ( )1 5.19 94 5.19 94 5.19 94 1 f x f x f x f x f x+ = + + + = +

( ) ( ) ( )1 18.94 89.19 18.94 89.19 f x f x f x+ = + +

( ) ( )18.94 89.19 1 f x f x + = + .

Vy ( ) ( )1 +1 f x f x+ = .Bi 3. Cho :f l hm kh vi cp hai vi o hm cp 2 dng.

Chng minh rng: ( )( ) ( )f x f x f x+ vi mi s thc x.Gii+ Nu ( ) 0f x = th ( )( ) ( )f x f x f x+ = vi mi x : hin nhin.

+ Nu ( ) 0f x < th p dng nh l Lagrange trn on ( );x f x x+ ta

c: ( ) ( )( ) ( ) ( )( )f x f x f x f c f x + = , ( )( );c x f x x + .


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( ) 0 f x f > l hm tng ( ) ( ) 0 f c f x < < . V vy

( ) ( )( ) 0f x f x f x + < .

+ Nu ( ) 0f x > th chng minh tng t nh trng hp ( ) 0f x < ta cng

thu c ( ) ( )( ) 0f x f x f x + < .

Bi 4 Cho 2x , chng minh ( )1 cos cos 11

x xx x

+ >

+.

Gii

Xt hm s: [ ): 2;f , ( ) cos f t t t

= .

p dng nh l Lagrange trn on [ ]; 1x x + i vi hm ( )f t

tn ti [ ] ( )( ) ( )

( )( ) ( )

1; 1 : 1

1

f x f xu x x f u f x f x

x x

+ + = = +

+

Cn chng minh ( ) [ )cos sin 1 u 2;f uu u u

= + > + .

( ) [ )2

3cos 0 u 2; f u f

u u

= < + nghch bin trn [ )2;+

( ) ( )lim 1u

f u f u

> = .

Vy ( )1 cos cos 11x x

x x + >+

[ )2;x + .

Bi 5 Tn ti hay khng hm kh vi lin tc f tha mn iu kin

( ) ( ) ( )2 , f f sin x f x x x x< ?

GiiKhng tn ti.Ta c:

( ) ( ) ( ) ( ) ( ) ( )2 2 20 0 0

0 2 2 sin 2 1 cos x x x

f x f f t dt f t f t dt tdt x = = =

Suy ra: ( ) ( ) ( )2 2 0 2 1 cos 4f f + .Bi 6

Gi s hm ( ) { ( ): ; \ 0 0; f a a + tho mn ( )( )01

lim 2x

f xf x

+ =

.

Chng minh rng ( )0

lim 1x

f x

= .

Gii


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Vi ( ) 0f x > , p dung bt ng thc Cauchy ta c: ( )( )

12f x

f x+ .

( )( )01

lim 2 0, 0x

f xf x

+ = > >

sao cho ( )

( )

10 2f x

f x + <

vi 0 x < < .

Ta c: ( )( )

( )( )( )

1 10 2 0 1 1 f x f x

f x f x

+ < +


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bt ng thc ( ) ( ) f x x c tho mn trong ln cn khuyt ca 0 v

( )0

lim 0x

x

= th t (*) suy ra c: ( )0

lim 0x

f x

= .

GiiV d

Xt :f xc nh bi ( )( )1

0

n

f x

=

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )2 2 2 2x f x f x f x f x f x f x x = + +

V ( ) ( ) ( ) ( )( )0

lim lim 2 0x x

x f x f x x

= + = nn ( )0

lim 0x

f x

= .

Bi 9a) Cho v d v hm f tho mn iu kin ( ) ( )( )

0lim 2 0x

f x f x

= nhng

( )0

limx

f x

khng tn ti.

b) Chng minh rng nu trong mt ln cn khuyt ca 0, cc bt ng thc

( )1

, 12

f x x

< < v ( ) ( )2 f x f x x c tho mn th ( )0

lim 0x

f x

= .

Gii

a) Xt :f

xc nh bi ( )

( )1

0

n

f x

=

b) ( )( )2 2

x x x f x

f x x

. Do1

12

< < nn ( )0

lim 0x

f x

= .

Bi 10

Cho trc s thc , gi s( )

( )limx

f axg a

x

= vi mi s dng a. Chng

minh rng tn ti c sao cho ( )g a ca= .

Gii

Ta c:( ) ( ) ( )

( ) ( ) ( )lim lim 1 1x t

g a f ax f t g g a g a

a a x t

= = = = . Chn

( )1c g= ta c ( )g a ca= .

Bi 11Gi s [ ]( )0;2f C v ( ) ( )0 2f f= . Chng minh rng tn ti 1 2, xx trong

[ ]0;2 sao cho 2 1 1x x = v ( ) ( )2 1 f x f x= .

nu1

, n = 0,1,2,3,...2n

x =

nu ngc li

nu ngc li

nu 1 , n = 0,1,2,3,...

2

nx =


5/35



Xt hm s ( ) ( ) ( )1g x f x f x= + , [ ]0;2x

V [ ]( )0;2f C nn [ ]( )0;2g C .

Ta c: ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )0 1 0 1 2 2 1 1g f f f f f f g= = = =

Suy ra: ( ) ( ) ( )2

0 1 1 0.g g g=

V th tn ti [ ] ( ) ( ) ( )0 0 0 00;1 : 0 1 x g x f x f x = + = .

Vy c th ly 2 0 1 01 , x x x x= + = .Bi 12Cho [ ]( )0;2f C . Chng minh rng tn ti 1 2,x x trong [ ]0;2 sao cho

2 11x x = v

( ) ( ) ( ) ( )( )2 11

2 02

f x f x f f = .

Gii

Xt hm s: ( ) ( ) ( ) ( ) ( )( )1

1 2 02

g x f x f x f f = + , [ ]0;2x

V [ ]( )0;2f C nn [ ]( )0;2g C .

Ta c: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )1 1

0 1 0 2 0 1 0 22 2

g f f f f f f f = = +

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )1 1

1 2 1 2 0 1 0 22 2g f f f f f f f

= = +

Suy ra: ( ) ( )0 1g g = ( ) ( ) ( )( )2

11 0 2 0

2 f f f

+

.

V th tn ti [ ] ( ) ( ) ( ) ( ) ( )( )0 0 0 01

0;1 : 0 1 2 02

x g x f x f x f f = + = .

Vy c th ly 2 0 1 01 , x x x x= + = .Bi 13Vi n , gi [ ]( )0; f C n sao cho ( ) ( )0 f f n= . Chng minh rng tn

ti 1 2;x x trong khong [ ]0;n tho mn 2 1 1x x = v ( ) ( )2 1 f x f x= .

GiiXt ( ) ( ) ( ) [ ]1 , x 0; 1g x f x f x n= +

( ) ( ) ( )0 1 ... 1g g g n+ + +

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 0 2 1 ... 1 0 0f f f f f n f n f n f = + + + = =

+ Nu ( ) 0g k = , { }0,1, 2,..., 1k n th ta c ngay iu phi chng minh.


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+ Nu {0,1, 2,..., 1k n : ( ) 0g k . Khng mt tnh tng qut gi s

( ) 0g k > th lc lun tm c { }, h 0,1,2,..., 1h k n sao cho

( ) 0g h < . Khi tn ti [ ]0 0; 1x n sao cho

( ) ( ) ( )0 0 00 1g x f x f x= + = .

Vy c th ly2 0 1 01 , x x x x= + = .

Bi 14Chng minh rng nu 1 2sin sin 2 ... sin sinna x a x a nx x+ + + vi x th

1 22 ... 1na a na+ + + .

Giit ( ) 1 2sin sin 2 ... sinn f x a x a x a nx= + + + ta c:

( )( ) ( )

1 2 0

02 ... 0 lim

nx

f x f a a na f

x

+ + + = =

( ) ( ) ( )0 0 0

sinlim lim . lim 1

sin sin x x x f x f x f xx

x x x x = = == .

Bi 15Gi s ( )0 0f = v f kh vi ti im 0. Hy tnh

( )01

lim ...2 3x x x x

f x f f f x k

+ + + +

vi k l mt s nguyn dng

cho trc.GiiTa c:

( )0

1lim ...

2 3x x x x

f x f f f x k

+ + + +

( ) ( )( ) ( ) ( )

0

0 0 00 1 1 12 3lim . . ... .

0 2 30 0 02 3x

x x x f f f f f f

f x f k

x x xx kk

= + + + +

= ( )( ) ( ) ( )

( )0 0 0 1 1 1

0 ... 1 ... 02 3 2 3

f f f f f

k k

+ + + + = + + + +

.


7/35



Bi 16Cho f l hm kh vi ti a v xt hai dy ( )nx v ( )ny cng hi t v a sao cho

n n x a y< < vi mi n . Chng minh rng: ( ) ( ) ( )lim n n

nn n

f x f yf a

x y

=

.

Gii

Ta c:( ) ( )

( )( ) ( ) ( ) ( )

0 n n n n n n

n n n n

f x f y f x f y x f a y f af a

x y x y

+ =

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( )( )

( ) ( )( ) ( )0

n n n n

n n

n n n n

n n n n

n n n n

n n n n

n n n n

n n

n n

n n

f x f y f a f a af a af a x f a y f a

x y

f x f a f a x a f y f a f a y a x y x y

f x f a f a x a f y f a f a y a

x y x y

f x f a f a x a f y f a f a y a

x a y a

f x f a f y f a f a f a n

x a y a

+ + +=

=

+

+

= +

Vy( ) ( )

( )lim n nn

n n

f x f yf a

x y

=

.

Bi 17Cho f kh vi trn ( )0;+ v 0a > . Chng minh rng:

a) Nu ( ) ( )( )limx

af x f x M +

+ = th ( )limx

Mf x

a+= .

b) Nu ( ) ( )

( )lim 2

x

af x x f x M +

+ = th ( )limx

Mf x

a+= .

Giip dng quy tc Lpitan, ta c:

a) ( )( ) ( )( )

( )

( ) ( )( )lim lim lim lim

ax axax

ax ax x x x xax

e f x e af x f xe f xf x

e aee+ + + +

+= = =

( ) ( )( ) ( ) ( )( )1 1

lim lim .x x

Maf x f x af x f x

a a a+ + = + = + =


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b) Ta c:

( )( ) ( )( )

( )lim lim lim

a xa x

a x x x xa x

e f xe f xf x

ee

+ + +

= =

( ) ( )2lim

2

a x

xa x

ae f x f xx

ae

x

+

+

=

( ) ( )( ) ( ) ( )( )1 1

lim 2 lim 2 .x x

Maf x x f x af x x f x

a a a+ + = + = + =

Cu 18Cho f kh vi cp 3 trn ( )0;+ . Liu t s tn ti ca gii hn

( ) ( ) ( ) ( )( )limx f x f x f x f x+ + + +

c suy ra s tn ti ca ( )limx f x+ khng?GiiKhng. Ly v d: ( ) ( )cos , x 0; f x x= + .Ta c:

( ) ( ) ( ) ( )( ) ( )lim lim cos sin cos sin 0x x

f x f x f x f x x x x x+ +

+ + + = + =

Nhng khng tn ti ( )lim lim cosx x

f x x+ +

= .

Cu 19

a) Gi s f xc nh v lin tc trn [ )0;+ , c o hm lin tc trn( )0;+ v tho mn ( )0 1f = , ( ) x 0x f x e . Chng minh rng tn ti

( )0 0;x + sao cho ( )0

0

x f x e

= .

b) Gi s f kh vi lin tc trn ( )1;+ v tho mn ( )1 1f = ,

( )1

x 1f xx

. Chng minh rng tn ti ( )0 1;x + sao cho

( )0 20

1f x

x = .

Giia) t ( ) ( ) xg x f x e=

f lin tc trn [ )0;+ g lin tc trn [ )0;+ g lin tc trn ti 0

( ) ( ) ( )0

lim 0 0 1 0x

g x g f +

= = = .

( ) ( )0 lim 0xx

f x e f x

+

=

( ) ( )( ) ( )lim lim lim lim 0x x x x x x

g x f x e f x e

+ + + +

= = = .


9/35



Do : ( ) ( ) ( ) ( )0 00

lim lim 0; : 0xx

g x g x x g x+ +

= + = hay ( ) 00x

f x e

= .

b) t ( ) ( )1

g x f x x= f kh vi lin tc trn ( ) ( ) ( )

11; lim 1 0

x f x f

+

+ = =

( ) ( )1 1

1lim lim 0x x

g x f xx

+ +

= =

.

( ) ( ) ( ) ( )1 1

0 lim 0 lim lim 0 x x x

f x f x g x f xx x+ + +

= = =

( ) ( ) ( ) ( )0 01

lim lim 1; : 0xx

g x g x x g x+ +

= + = hay ( )0 20

1f x

x

= .

Cu 20 Cho [ ]( ) ( ) ( )0 0

0;1 : sin cos 1 M f C f x xdx f x xdx

= = =

.

Tm ( )20

minf M

f x dx

.

Gii

Cho ( ) ( )02

sin cos f x x x

= + .

+ R rng 0f M .

+ i vi hm bt k f M , ( ) ( )2

00

0 f x f x dx

.

Suy ra: ( ) ( ) ( ) ( ) ( )2 2 20 0 00 0 0 0

8 4 42 f x dx f x f x dx f x dx f x dx

= = = .

Vy cc tiu t c khi0f f= .

Cu 21Tm hm s ( )f x c o hm lin tc trn sao cho

( ) ( ) ( )( )2 2 2

0 2011

x

f x f t f t dt = + +

(1).GiiV hm s ( )f x c o hm lin tc trn nn ( )2f x c o hm lin tctrn .Ly o hm 2 v ca (1), ta c:

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )22 22 0f x f x f x f x f x f x f x f x = + = =

( ) x f x Ce = (2).


10/35



T (1) suy ra: ( ) ( )2 0 2011 0 2011f f= = .

Cho 0x = , t( ) ( )2 0 2011f C = = .

Vy ( ) 2011 x f x e= .

Cu 22Tm tt c cc hm s lin tc :f tho mn

( ) ( ) ( ) ( ) ( ) ( )1 2 2011 1 2 2011... ...f x f x f x f y f y f y+ + + = + + +

vi mi b s tho mn:1 2 2011 1 2 2011

... ... 0 x x x y y y+ + + = + + + = .Giit ( ) ( ) ( )0 , g . f b x f x b= = Do : ( ) ( )0 0 0g f b= =

v ( ) ( ) ( ) ( ) ( ) ( )1 2 2011 1 2 2011... ...g x g x g x g y g y g y+ + + = + + +

vi mi b s tho mn : 1 2 2011 1 2 2011... ... 0 x x x y y y+ + + = + + + = .Trc ht cho

1 2 2011 1 2 2009 2010 2011... 0 , x ... 0 , x , x y y y x x x x= = = = = = = = = =

ta c: ( ) ( ) xg x g x = .

Tip theo cho

1 2 2011 1 2 2008 2009 2010 2011... 0 , x ... 0 , x , x , y y y x x x y x x y= = = = = = = = = = = ta c:

( ) ( ) ( ) ( ) ( ) ( )0 x,y x, yg x g y g x y g x y g x g y+ + = + = +

y l phng trnh hm Cauchy, do : ( )g x ax= , ( )1a g= .

Vy ( ) , a, b = const f x ax b= + .

Cu 23Cho f lin tc trn on [ ];a b , kh vi trong khong ( );a b v

( ) ( ) 0 f a f b= = . Chng minh rng tn ti ( );c a b sao cho:

( ) ( )2011 f c f c = .

GiiXt hm s: ( )

( )

( )2010

x

a

f t dt

g x e f x

=

V f lin tc trn on [ ];a b , kh vi trong khong ( );a b nn g lin tc trn

on [ ];a b , kh vi trong khong ( );a b . Hn na ( ) ( ) 0g a g b= = suy ra tn

ti ( ) ( ); : 0c a b g c = .

M ( )( )

( ) ( )( )2010

2011

x

a

f t dt

g x e f x f x

= . Suy ra: ( ) ( )2011 f c f c = .


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Cu 24Cho f lin tc trn [ ]0;2012 . Chng minh rng tn ti cc s

[ ]1 2 1 2, 0;2012 , x 1006 x x x = tho mn: ( ) ( )( ) ( )

2 12012 0

2f f

f x f x

=

Gii

Xt hm s: ( )( ) ( ) ( ) ( )1006 2012 0

1006 2012

x f x f f F x

+ = , [ ]0;1006x .

F lin tc trn [ ]0;1006 . Ta c:

( )( ) ( ) ( )

( ) ( ) ( ) ( )

2 1006 2012 00

2012

2 1006 2012 010062012

f f f F

f f f F

=

=

( ) ( ) [ ] ( )0 00 1006 0 0;1006 : 0F F x F x = .

[ ] ( ) ( )( ) ( )

0 0 0

2012 00;1006 : 1006

2

f f x f x f x

+ = .

t 2 0 1 01006 , x x x x= + = ta c iu phi chng minh.Cu 25Cho s thc a [ ]0;1 . Xc nh tt c cc hm lin tc khng m trn [ ]0;1

sao cho cc iu kin sau y c tha mn:a) ( )

1

0

1 f x dx = b) ( )1

0

xf x dx a= c) ( )1

2 2

0

x f x dx a= .

Giip dng bt ng thc Bunhiacovski ta c:

( ) ( ) ( ) ( ) ( )2 21 1 1 1

2

0 0 0 0

. . xf x dx x f x f x dx x f x dx f x dx =

.

M theo gi thit: ( ) ( ) ( )21 1 1

2

0 0 0

.xf x dx x f x dx f x dx =

.

Do f lin tc trn [ ]0;1 nn ( ) ( ) [ ]0, x 0;1 x f x f x =

Suy ra: ( ) [ ]0 x 0;1f x = . iu ny mu thun vi gi thit: ( )1

0

1 f x dx = .

Vy khng tn ti hm f tho mn bi ton.Bi 26C tn ti hay khng hm s kh vi :f tho mn

( ) ( ) ( )20 1 , f x ? f x f x=


12/35



GiiGi s hm f tho mn yu cu bi ton. V ( ) ( )2 0 x f x f x nn

fng bin trn [ ) ( ) ( ) [ )0; 0 1 0 x 0; f x f + = > + .

T gi thit bi ton ta c:( )

( )( ) [ )2

0 0

1, x 0;1

1

x xf tdt dt f x

f t x

.

Do khng tn ti ( )1

limx

f x

. iu ny mu thun vi gi thit f lin tc.

Vy khng tn ti hm f tho mn bi ton.Cu 27C hay khng mt hm s :f tha mn: ( ) sin sin 2 f x y x y+ + + <

vi x, y .GiiGii s tn ti hm f tho mn yu cu bi ton.

+ Cho , y =2 2

x

= , ta c: ( ) 2 2f + < .

+ Cho3

, y =2 2

x

= , ta c: ( ) 2 2f < .

Ta li c: ( )( ) ( )( ) ( ) ( )4 2 2 2 2 4 f f f f = + + + + + < . iu

ny v l. Vy khng c hm s f no tho yu cu bi ton.

Cu 28Tm tt c cc hm f(x) xc nh v lin tc trn sao cho( ) ( ) 0 x f x f x = .

Gii

t ( ) ( )( )2

g x f x=

( ) ( ) ( )2 0 xg x f x f x = =

( ) ( ) ( )g x C const f x const f x ax b = = = = + x .

Cu 29

Cho :f sao cho ( ) ( ) a bf a f b a b < . Chng minh rngnu ( )( )( )0 0 f f f = th ( )0 0f = .GiiTa vit li iu kin i vi hm f(x) nh sau: ( ) ( )f a f b a b (*)

Du = xy ra khi a = b.t ( ) ( )0 , y = f x f x= . Khi ( ) 0.f y =

p dng bt ng thc ( )* lin tip ta c:


13/35



( ) ( ) ( ) ( ) ( ) ( )0 0 0 0x x f x f y x f y f x y f f y x= = = =

Suy ra: 0x y= = . Vy ( )0 0f = .

Cu 30

Hm ( )2

3 12

x x f x e x= c kh vi ti im 0x = hay khng?

GiiTheo cng thc Taylor, ta c:

( ) ( )2 3 2 3

3 31 12 6 2 6

x x x x x xe x o x e x o x= + + + + = +

( ) ( ) ( )

333

3

1

6 6

x

f x o x x o x= + = +

.

Vy f(x) kh vi ti 0x = v ( )3

10

6f = .

Cu 31Chng minh rng nu hm f(x) kh vi v hn ln trn th hm

( ) ( )0 f x f

x

c nh ngha thm lin tc ti x = 0 cng kh vi v hn

ln.

GiiVi 0x ta c:

( ) ( ) ( ) ( )( ) ( )

( )1 1

0 0 0

00

x f x f f x f f t dt f ux xdu f ux du

x

= = =

V ( )1

0

f ux du kh vi v hn ln vi mi x .

Vy( ) ( )0 f x f

x

c nh ngha thm lin tc ti x = 0 kh vi v hn

ln.

Cu 32Cho ( )f x kh vi 2 ln tho ( ) ( )0 1 0f f= = ,

[ ]( )

0;1in 1

xm f x

= .

Chng minh rng:[ ]

( )0;1

max 8x

f x

.

Giif lin tc trn [ ] [ ] ( )

[ ]( )

0;10;1 0;1 : in 1

xa f a m f x

= = .Suy ra c

( ) 0f a = , ( )0;1a .


14/35



Khai trin Taylor ti a: ( )( )( )

( )2

12

f a x a f x x a

+ = + , 0 1< < .

+ Vi 0x = , ta c: ( ) 210 12

f ca

= + , 10 c a< < .

+ Vi 1x = , ta c:( )

( )220 1 1

2

f ca

= + , 2 1a c< < .

Do : ( )1 22

8f ca

= nu1

2

a ; ( )( )

22

28

1f c

a =

nu

1

2a .

Vy[ ]

( )0;1

max 8x

f x

.

Cu 33

Gi s ( )2011 1sin , x 0

0 , x = 0

xf x x

=

v hm ( )g x kh vi ti x = 0. Chng minh rng ( )( )g f x c o hm bng0 ti 0x = .Gii

Ta c: ( )( )( )( ) ( )( ) ( )

2011

0 00

1sin 00

lim limh hx

g h gg f h g f d h

g f xdx h h =

= =

( ) ( )2011 20112011 2011

0 0 02011 2011

1 1sin 0 sin 0

1 1lim . sin lim . lim sin

1 1sin 0 sin 0

h h h

g h g g h gh h

h hh h

h hh h

= =

V ( )2011 20111

0 sin 0 0h h hh

nn 20110

1lim sin 0h

hh

= .

Do : ( )( ) ( )0

0 .0 0x

dg f x g

dx=

= =

Cu 34Hm f xc nh, kh vi trn ( )0; , + . Chng minh rng hm

( ) ( ) f x f x + khng gim khi v ch khi ( ) x f x e khng gim.

Giit ( ) ( ) ( )h x f x f x= + ; ( ) ( ) xg x f x e= .


15/35



Suy ra: ( ) ( )( )x xe h x e f x = ; ( ) ( )xe g x f x = .

Khi :( ) ( ) ( ) ( ) ( ) ( )( ) ( )

0

0x

x x t g x e f x h x e f x h x e f t dt f

= = =

( ) ( ) ( )0

0x

th x e h t dt f

= .

( ) ( ) ( ) ( ) ( ) ( )0

0x

xh x f x f x e g x f t dt f

= + = + +

= ( ) ( ) ( )0

0x

x te g x e g t dt f

+ + .

( ) Gi s ( )h x khng gimKhi vi b > a ta c:

( ) ( ) ( ) ( )( ) ( )b

b a t

a

g b g a e h b e h a e h t dt = (1)

Theo nh l trung bnh ca tch phn tn ti

( ) ( ) ( ) ( )( )1

; :b b

t t b a

a a

c a b e h t dt h c e dt h c e e

= = (2)

Thay (2) vo (1) ta c:

( ) ( ) ( ) ( ) ( ) ( )b a b a

g b g a e h b e h a e h c e h c

= + ( ) ( )( ) ( ) ( )( ) 0b ae h b h c e h c h a = + vi b c a> > .

Do g(x) khng gim.( ) Gi s g(x) khng gim

Khi vi b > a ta c:

( ) ( ) ( ) ( )( ) ( )b

b a t

a

h b h a e g b e g a e g t dt = + (3)

Theo nh l trung bnh ca tch phn tn ti

( ) ( ) ( ) ( )( )1; :b b

t t b a

a a

c a b e g t dt g c e dt g c e e

= = (4)Thay (4) vo (3) ta c:

( ) ( ) ( ) ( ) ( ) ( )b a b ah b h a e g b e g a e g c e g c = +

( ) ( )( ) ( ) ( )( ) 0b ae g b g c e g c g a = + vi b c a> > .Do h(x) khng gim.Vy bi ton chng minh xong.


16/35



Cu 35Gi s ( )f C . Liu c tn ti cc hm s g(x) v h(x) sao cho x

th ( ) ( ) ( )sin cosf x g x x h x x= + hay khng?GiiC. Chng hn xt cc hm s sau:

( ) ( ) ( ) ( )sin , h cosg x f x x x f x x= =

Ta c: ( ) ( ) ( ) ( ) ( )2 2sin cos sin cosg x x h x x f x x f x x f x+ = + = .

Cu 36Gi s :f c o hm cp 2 tho mn: ( ) ( )0 1, f 0 0f = = v

( ) ( ) ( ) [ )5 6 0 0; f x f x f x x + + . Chng minh rng:

( ) 2 33 2x x f x e e , [ )0;x + .GiiTa c:

( ) ( ) ( ) [ )5 6 0 0; f x f x f x x + +

( ) ( ) ( ) ( )( ) [ )2 3 2 0 0;f x f x f x f x x +

t ( ) ( ) ( ) [ )2 , x 0;g x f x f x= + .

Khi ( ) ( ) [ ) ( )( ) [ )33 0 , x 0; 0 ,x 0;xg x g x e g x + +

( )3x

e g x

tng trn [ )0;+ ( )( ) [ ) ( )( ) [ )2 22 , x 0; 2 0 x 0; x x x xe f x e e f x e + + +

( )2 2x xe f x e + tng trn [ )0;+

( ) ( ) [ )2 0 02 0 2 3 , 0;x xe f x e e f e + + = +

( ) 2 33 2x x f x e e , [ )0;x + .

Cu 37Cho ( ): 0;f + c o hm cp 2 lin tc tho mn:

( ) ( ) ( ) ( )2

2 1 2011 f x xf x x f x + + + vi mi x. Chng minh rng:( )lim 0

xf x

= .

Giip dng quy tc Lpitan, ta c:

( )( )

2

2

2

2

lim lim

x

xx x

e f xf x

e

= =


17/35



=

( )

( ) ( )( )

( ) ( )( )2 2

2

2

2 2

2 22

22 2

lim lim lim

x x

x

x x x xx x

e f x e f x xf xe f x xf x

xee xe

+ +

= =

( ) ( ) ( ) ( )( )

( )

( ) ( ) ( ) ( )2

2

2 22

222

2 1 2 1lim lim 0.

11

x

xx x

e f x xf x x f x f x xf x x f x

xe x

+ + + + + += = =

++

Cu 38

Gi s hm s f lin tc trn [ ) ( )0; , 0 0 f x x+ v ( )lim 1x

f xa

x+= < .

Chng minh rng tn ti 0c sao cho ( ) f c c= .

Gii+ Nu ( )0 0f = th kt lun trn hon ton ng.

+ Nu ( )0 0f >

t ( ) ( )g x f x x=

V f lin tc trn [ )0;+ g cng lin tc trn [ )0;+ .

Ta c: ( ) ( ) ( )0 0 0 0 0 x 0g f f= = > .

( ) ( )( )lim 1 0 : 1 0 :

x

f x f ba b b f b b

x b+= < > < > < .

Khi : ( ) ( ) 0g b f b b= < .

( ) ( ) [ ] [ ) ( ) ( )0 0 0; 0; : 0 0 :g g b c b g c c f c c + = = .Cu 39Gi s f c o hm trn mt khong cha [ ]0,1 , ( ) ( )0 0 , f 1 0f > < .

Chng minh rng tn ti ( ) ( ) ( ) [ ]0 00;1 : x 0;1 x f x f x

.Giif c o hm trn mt khong cha [ ]0,1

[ ] ( ) ( )[ ]

( )0 0 0,10;1 : maxx x f x f x f x = .

Ta s chng minh: 0 00, x 1x .Tht vy!

( ) ( )( ) ( )

( ) ( )( ]

0

0 0lim 0 0 0;1 : 0 x 0;x

f x f f x f f h h

x x+

= > >


18/35



( ) ( ) ( ] ( )0 x 0; 0 f x f h f > khng phi l gi tr ln nht ca ( )f x

trn [ ] 00,1 0x .

( ) ( )( ) ( )

( ) ( )[ )

11

1 1lim 1 0 0;1 : 0 x ;1

1 1x f x f f x f

f k k x x

= < <

( ) ( ) [ ) ( )1 x ;1 1 f x f k f < khng phi l gi tr ln nht ca ( )f x

trn [ ) 0;1 1k x .

Cu 40Cho mt hm s f xc nh trn tho mn

( ) ( )0 0 , f sin x f x x= . Chng minh rng o hm ca f ti 0

khng tn ti.

GiiGi s ( )0f tn ti.

0;2

x

ta c:

( ) ( )( )

( ) ( )0 0

0 0sin sin0 lim lim 1

0 0x x f x f f x f x x

f x x x x

+ +

+

= =

.

Tng t ta cng chng minh c ( )10 1f <

iu ny chng t ( )0f khng tn ti.

Cu 41Gi s ( )f x kh vi trn ( );a b sao cho ( )lim , lim

x a x bf x

+

= + = v

( ) ( ) ( )2 1 x ; f x f x a b + . Chng minh rng b a . Cho v d

b a = .GiiCch 1

Ta c: ( ) ( ) ( )( )

( )( )2 21 x ; 1 0 x ;1

f x f x f x a b a b

f x

+ +

+

( )( ) ( ) ( )arctan 0 x ; arctan f x x a b f x x + + tng trn ( );a b

Chuyn qua gii hn ta c:2 2

a b b a

+ + .

V d: cot , a = 0 , b =y x = .Cch 2

Ta c: ( ) ( ) ( )( )

( )( )2 21 x ; 1 x ;1

f x f x f x a b a b

f x

+

+


19/35



Ly tch phn hai v:( )

( ) ( )21 arctan

1

b bb

aa a

f xdx dx f x a b a b b a

f x

+ .

Cu 42

Cho f l mt hm lin tc trn [ ]0;1 . Tm . Tm ( )1

0

lim nn

f x dx

.

Gii

Cho 0 1< < . Khi ta c: ( ) ( ) ( )1 1 1

0 0 1

n n n f x dx f x dx f x dx

= + .

+ Theo nh l gi tr trung bnh ca tch phn tn ti

[ ] ( ) ( )( ) ( ) ( )( )1 1

0 00;1 : 1 lim 0 1n n nnc f x dx f c f x dx f

= = .

+ t[ ]

( )0,1

supx

M f x

= , ta c: ( ) ( )1 1

1 1

n n f x dx f x dx M

.

Vy ( ) ( )1

0

lim 0nn

f x dx f

= .

Cu 43

Cho f l mt hm lin tc trn [ ];a b v ( ) 0b

a

f x dx = . Chng minh rng tn

ti ( ) ( ) ( ); :c

a

c a b f x dx f c = .

Xt hm: ( ) ( )x

x

a

g x e f t dt

=

g lin tc trn [ ];a b , kh vi trn ( );a b

( ) ( ) 0g a g b= = .

Theo nh l Rolle tn ti ( ) ( ); : 0c a b g c = .

M ( ) ( ) ( )

x

x

ag x e f x f t dt

= , v th ( ) ( ) ( )

c c

a a f c f t dt f x dx= = .

Cu 44

Gi s [ ]( );f C a b , a > 0 v ( ) 0b

a

f x dx = . Chng minh tn ti ( );c a b

sao cho ( ) ( )c

a

f x dx cf c= .


20/35



Gii

Xt hm s: ( ) ( )1 x

a

g x f t dt x

=

g lin tc trn [ ];a b , kh vi trn ( );a b

( ) ( ) 0g a g b= = .


M ( ) ( ) ( )21 x

a

g x xf x f t dt x

=

Do tn ti ( );c a b sao cho ( ) ( )c

a

f x dx cf c= .

Cu 45Gi s f, g [ ]( );C a b . Chng minh rng tn ti ( );c a b sao cho

( ) ( ) ( ) ( )b b

a a

g c f x dx f c f x dx= .

Gii

Xt ( ) ( ) ( ) ( ), Gx x

a a

F x f t dt x g t dt = =

Suy ra: ( ) ( )F x f x = , ( ) ( )G x g x =

p dng nh l Cauhy ta c:

c ( );a b :( ) ( )

( ) ( )

( )

( )

F b F a F c

G b G a G c

=

c ( );a b :

( )

( )

( )

( )

b

a

b

a

f t dt f c

g cg t dt

=

c ( );a b : ( ) ( ) ( ) ( )b b

a a


Cu 46

Gi s f, g [ ]( );C a b

. Chng minh rng tn ti ( );c a b

sao cho( ) ( ) ( ) ( )

c b

a c


Gii

Xt hm: ( ) ( ) ( )x b

a x

F x f t dt g t dt =

F lin tc trn [ ];a b , kh vi trn ( );a b v ( ) ( )F a F b= .

V th theo nh l Rolle ta c: ( ) ( ); : 0c a b F c =


21/35

www.MATHVN.com- N THI OLYMPIC TON SINH VIN TON QUC phn GII TCH


M ( ) ( ) ( ) ( ) ( )b x

x a

F x f x g t dt g x f t dt =

Do : ( ); :c a b ( ) ( ) ( ) ( )c b

a c


Cu 47Gi s f v g l hai hm s dng, lin tc trn [ ];a b . Chng minh rng tn

ti ( );c a b sao cho( )

( )

( )

( )1

c b

a c

f c g c

f x dx g x dx

=

.

Gii

Xt hm: ( ) ( ) ( )x b

x

a x

F x e f t dt g t dt =

F lin tc trn [ ];a b , kh vi trn ( );a b v ( ) ( )F a F b= .

Theo nh l Rolle ta c: ( );c a b : ( ) 0F c = .

M: ( ) ( ) ( ) ( ) ( ) ( ) ( ) x b b x

x

a x x a

F x e f t dx g t dx f x g t dt g x f t dt = +

Do : ( );c a b : ( ) ( ) ( ) ( ) ( ) ( ) 0c b b c

a c c a

f t dx g t dx f x g t dt g x f t dt + =

( );c a b : ( )

( )

( )

( )1

c b

a c

f c g c

f x dx g x dx

=

.

Cu 48Cho [ ]( )1 0;1f C . Chng minh rng tn ti ( )0;1c sao cho:

( ) ( ) ( )1

0

10

2 f x dx f f c= + .

Gii

Ta c: ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1

10

0 0 01 1 1 f x dx f x d x x f x x f x dx= =

( ) ( ) ( )1

0

0 1 f x f x dx= .

Theo nh l gi tr trung bnh ca tch phn:

tn ti ( ) ( ) ( ) ( ) ( ) ( )1 1

0 0

10;1 : 1 1

2c x f x dx f c x dx f c = = .


22/35



Do : tn ti ( )0;1c sao cho: ( ) ( ) ( )1

0

10

2 f x dx f f c= +

Cu 49Cho [ ]( )2 0;1f C . Chng minh rng tn ti ( )0;1c sao cho:

( ) ( ) ( ) ( )1

0

1 10 0

2 6 f x dx f f f c = + + .

Gii

Ta c: ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1

1

00 0 0

1 1 1 f x dx f x d x x f x x f x dx= =

( )

( )

( )

( )

( )

12 21

00

1 1

0 2 2

x x

f f x f x dx

= + .p dng nh l gi tr trung bnh ca tch phn:

tn ti ( )( )

( ) ( ) ( ) ( )2

1 12

0 0

1 1 10;1 : 1

2 2 6

xc f x dx f c x dx f c

= = .

Do tn ti ( )0;1c sao cho: ( ) ( ) ( ) ( )1

0

1 10 0

2 6 f x dx f f f c = + + .

Cu 50

Gi s [ ]( )1 0;1f C v ( )0 0f . Vi ( ]0;1x , cho ( )x tho mn

( ) ( )( )0

x

f t dt f x x= . Tm( )

0limx

x

x

+

.

Gii

t ( ) ( )0

x

F x f t dt = .

Suy ra: ( )0 0F = , ( ) ( ) ( ) ( ), FF x f x x f x = = .

Ta c: ( ) ( )0 0 0F f = .

Theo khai trin Taylor ta c: ( ) ( ) ( ) ( )2 210 02

F x F x F x o x = + +

( ) ( ) ( ) ( )0 0F x F F x o x = + + ( ) ( ) ( ) ( )0 0F F F o = + +

( )( ) ( ) ( ) ( ) ( )0 0 f x x F x x F F o = = + +

Khi : ( ) ( ) ( )2 21

0 02

F x F x o x + + = ( ) ( ) ( )0 0 x F F o + +


23/35



( )

0

1lim

2xx

x

+

= .

Cu 51Cho f l mt hm lin tc trn v a b< , k hiu

( ) ( )2011b

a

g x f x t dt = + . Tnh o hm ca g.

Gii

Ta c: ( ) ( ) ( )2011

2011

2011b b x

a a x

g x f x t dt f u du+

+

= + =

( ) ( ) ( )2011 2011 2011g x f b x f a x = + + .

Cu 52Cho f lin tc trn . Tm ( ) ( )( )

0

1lim

b

ha

f x h f x dxh

+ .

Giip dng nh l gi tr trung bnh ca tch phn, ta c:

( ) ( )( ) ( ) ( )b b h b

a a h a

f x h f x dx f x dx f x dx+

+

+ =

( ) ( ) ( ) ( )b b h a h b

a h b a a h

f x dx f x dx f x dx f x dx+ +

+ +

= +

, [ ], 0,1 .

( ) ( )( ) ( ) ( )0

1lim

b

ha

f x h f x dx f b f ah

+ = .

Cu 53

Cho f l mt hm lin tc trn [ )0;+ tho mn ( ) ( )0

limx

x f x f t dt

+

c

gii hn hu hn. Chng minh ( )lim 0x f x = .Gii

t ( ) ( ) ( ) ( )0

x

F x f t dt F x f x= = .

Khi gi s ( ) ( ) ( ) ( )( )0

lim limx

x x f x f t dt F x F x L

+ = + =

( ) ( ) ( ) ( )a b h

a h b

f x dx f x dx hf a h hf b h +

+

= + = + + +


24/35



p dng quy tc Lpitan ta c:

( )( ) ( )( )

( )

( ) ( )( )( ) ( )( )lim lim lim lim lim

x xx

x x x x x x xx

e F x e F x F xe F xF x F x F x Le ee

+

= = = = + =

Suy ra: ( ) ( )lim lim 0x x

f x F x

= = .

Cu 54Chng minh rng nu f kh tch Riemann trn [ ];a b th

( ) ( ) ( ) ( )2 2

2sin cosb b b

a a a

f x xdx f x xdx b a f x dx

+

.

Gii

p dng bt ng thc Schwarz, ta c:

( ) ( )

( ) ( ) ( ) ( )

2 2

2 2 2 2 2

sin cos

sin cos

b b

a a

b b b b b

a a a a a

f x xdx f x xdx

f x dx xdx f x dx xdx b a f x dx

+

+ =

Cu 55Chng minh rng nu f dng v kh tch Riemann trn [ ];a b th

( ) ( ) ( )

2b b

a a

dx

b a f x dx f x .

Hn na nu ( )0 m f x M < th ( )( )

( )( )

22

4

b b

a a

m Mdx f x dx b a

f x mM

+ .

Gii+ p dng bt ng thc Cauchy Schwarz, ta c:

( ) ( )( )

( )( )

2

2 1.

b b b

a a a

dxb a f x dx f x dx

f xf x

= .

+ V ( )0 m f x M < nn ( )( ) ( )( )( )

0 , a x b f x m f x M

f x

Ta c:

( )( ) ( )( )( )

( ) ( )( )

0 0b b b b

a a a a

f x m f x M dxdx f x dx m M dx mM

f x f x

+ +


25/35



( )( )

( )( )( )

( )( ) ( ) .b b b b

a a a a

dx dx f x dx mM m M b a mM m M b a f x dx

f x f x + + +

Do : ( )( )

( )( ) ( ) ( )2b b b b

a a a a

dxmM f x dx m M b a f x dx f x dx

f x

+

Xt hm s: ( ) 2 y g t t kt = = + .

Hm st cc i ti2

kt= vi gi tr cc i l

2

4

k.

Vi ( )( ) ( ), t =b

a

k m M b a f x dx= + ta c:

( )( ) ( ) ( ) ( ) ( )

2 22

4

b b

a a

m M b am M b a f x dx f x dx

+ + .

Do : ( )( )

b b

a a

dxmM f x dx

f x

( ) ( )2 2

4

m M b a+

( )( )

b b

a a

dx f x dx

f x

( ) ( )2 2

4

m M b a

mM

+ .

Cu 56Cho f lin tc trn [ ];a b sao cho vi mi [ ] [ ]; ;a b ta c:

( )1

f x dx M

+

vi 0 , >0M > .

Chng minh rng ( ) 0f x = trn [ ];a b .

GiiVi mi [ ]0 ; x a b , chn h thuc b sao cho [ ]0 ; x h a b+ .Khi theo nh l trung bnh ca tch phn: tn ti c gia 0x v 0x h+

sao cho ( ) ( ) ( )0

0

1x h

x

f c h f x dx h f c M h

++

= .

Cho 0h ta c ( ) [ ]0 00 x ; f x a b . Suy ra: ( ) 0f x = trn [ ];a b .

Cu 57

Cho f lin tc trn [ ];a b . t ( )1 b

a

c f x dxb a

=

. Chng minh rng:

( ) ( )2 2

b b

a a

f x c dx f x t dx t .


26/35



Gii

Xt ( ) ( ) ( ) ( )2 2 22

b b b

a a a

g t x t dt b a t f x dx t f x dx

= = +

.

g(t) l tam thc bc hai theo t, g(t) t cc tiu ti ( )01 b

a

t f x dx cb a

= =

.

Vy ( ) ( )2 2

b b

a a

f x c dx f x t dx t .

Cu 58Cho f l mt hm thc kh vi n cp 1n + trn . Chng minh rng vi

mi s thc , , a < ba b tho mn( ) ( ) ( ) ( )

( ) ( )( )

( )

...ln

...

n

n

f b f b f bb a

f a f a f a

+ + +=

+ + +

tn ti ( );c a b sao cho ( ) ( ) ( )1n

f c f c+

= .

GiiVi a, b l s thc, a b< ta c

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

...ln

...

n

n

f b f b f bb a

f a f a f a

+ + +=

+ + +

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )... ...n na bf a f a f a e f b f b f b e + + + = + + +

Xt hm s

: ( ) ( ) ( )

( )

( )( )...n x

g x f x f x f x e

= + + +

Ta c g(x) kh vi trn v ( ) ( )g a g b= .


M ( ) ( ) ( ) ( )( )1nxg x e f x f x+ = .

Do : ( ) ( ) ( )1n

f c f c+

= .

Cu 59Cho [ ): 0;f + l mt hm lin tc kh vi. Chng minh rng:

( ) ( ) ( ) [ ] ( ) ( )

21 1 1

3 20,1

0 0 00 maxx f x dx f f x dx f x f x dx

.

Giit

[ ]( )

0,1maxx

M f x

= .

Khi ( ) [ ] ( ) [ ]x 0;1 x 0;1 f x M M f x M . Nhn ( ) 0f x

vo tng v ca bt ng thc ny ta c :( ) ( ) ( ) ( ) Mf x f x f x Mf x , [ ]0;1x


27/35



Suy ra: ( ) ( ) ( ) ( )0 0 0

x x x

M f t dt f t f t dt M f t dt

( ) ( ) ( ) ( )2 20 0

1 1 02 2

x x

M f t dt f x f M f t dt . n y ta tip tc nhn

( ) 0f x vo tng v ca bt ng thc ny c:

( ) ( ) ( ) ( ) ( ) ( ) ( )3 20 0

1 10

2 2

x x

Mf x f t dt f x f f x Mf x f t dt , [ ]0;1x .

Ly tch phn 2 v trn [ ]0;1 ca bt ng thc ny:

( )21

0

M f x dx

( ) ( ) ( ) ( )21 1 1

3 2

0 0 0

0 f x dx f f x dx M f x dx

( ) ( ) ( ) ( )21 1 1

2 2

0 0 0

0 f x dx f f x dx M f x dx

hay ( ) ( ) ( )[ ]

( ) ( )1 1 1

3 2

0,10 0 0

0 maxx

f x dx f f x dx f x f x dx

.

Cu 60

Cho [ ): 0;f + kh vi v tho mn ( ) ( )( )2 2

11 1 , f f x

x f x= =

+.

Chng minh rng tn ti gii hn hu hn ( )limx f x+ v b thua 1 4

+ .

Gii

( )( )

[ )2 21

0 x 0;f x x f x

= > ++

f(x) ngbin ( ) ( )1 1 x > 1 f x f > = .

T ta c: ( ) ( ) 2 11 1

11 1 arctan 1

1 4

x xx

f x f t dt dt t t

= + < = + < +

+ .

Vy tn ti gii hn hu hn ( )limx f x+ v b thua 1 4

+ .Cu 61Tm tt c cc hm ( )f x tho mn iu kin: ( ) ( )1 2 x f x f x + = .

GiiNhn xt: ( )1 112 2.2 2 2 2 x x x x x + = = Ta c: ( ) ( )1 2 x f x f x + = ( ) ( ) ( )1 1 11 2 2 x

x x f x f x

+ + + = +

t ( ) ( ) 12 xg x f x = + ( ) ( )1 xg x g x+ = . Vy ( ) ( ) 12 x f x g x = ,

vi g l hm tun hon c chu k 1T = .


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Cu 62Cho f l hm lin tc trn [ )0;+ v tho mn ( )0 3 1 xf x< < ( )0;x + .

Chng minh rng hm s ( ) ( ) ( )3

3

0 0

3x x

g x t f t dt tf t dt =

l hm sng

bin trn ( )0;+ .

Gii

Ta c: ( ) ( ) ( ) ( ) ( ) ( )2 2

3 2

0 0

9 3x x

g x x f x xf x tf t dt xf x x tf t dt

= =

Li c: ( ) ( ) ( )2 2

2 2

0 0 0 0

0 3 1 3 3 0 x x x x

tf t dt dt x tf t dt x x tf t dt

< < = < >

Kt hp vi ( ) ( )0 0; xf x x> + , ta suy ra: ( ) ( )0 x 0;g x > + .

Vy ( )g x l hm sng bin trn ( )0;+ .

Cu 63Cho hm s: [ ]( )2 0,2f C v ( ) ( ) ( )0 2010, f 1 2011, f 2 2012f = = = .

Chng minh rng tn ti ( )0;2c sao cho ( ) 0f c = .

Gii+ p dng nh l Lagrange cho hm s f trn [ ] [ ]0;1 , 1;2

( ) ( ) ( ) ( )1 0 2011 20100;2 : 11 0 1 0

f fa f a = = =

( ) ( )( ) ( )2 1 2012 2011

0;2 : 12 1 2 1

f fb f b

= = =

+ V f kh vi trn [ ]0;2 v ( ) ( ) f a f b = nn theo nh l Rolle tn ti

( ) ( )0;2 : 0c f c = .

Cu 64Tn ti hay khng hm lin tc :f + + tho mn cc iu kin:

( ) ( )( ) f 2011 2012i f< ( ) ( )( ) 1fii f xx

= .

Gii+ Trc ht ta chng minh f l n nh.

1 2,x x+ , ta c:

( ) ( ) ( )( ) ( )( )1 2 1 2 1 21 2

1 1f x f x f f x f f x x x

x x= = = = .


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+ f lin tc v n nh suy ra fn iu. Kt hp vi iu kin (i) suy ra f

ng bin trn + . Khi ( )( )1

f f xx

= cng l hm ng bin. iu ny

v l v1

yx

= l hm nghch bin.

Vy khng tn ti hm f tho mn yu cu bi ton.Cu 65Cho f xc nh trn [ ]0;1 tho mn: ( ) ( )0 1 0f f= = v

( ) ( ) [ ]x, y 0,12

x y f f x f y

+ +

.

Chng minh rng: phng trnh ( ) 0f x = c v s nghim trn on [ ]0,1 .GiiCho x y= , t gi thit ta c: ( ) ( ) ( ) [ ]2 0 x 0,1 f x f x f x .

Ta c: ( ) ( )1 1

0 0 1 0 02 2

f f f f

+ = =

.

Ta s chng minh1

02n

f

=

n (1)

+ (1) ng vi 0, 1n n= = .

+ Gi s (1) ng n n k= , tc l: 1 02kf =

.

+ Ta c: ( )1 11 1 1

0 0 0 02 2 2k k k

f f f f + +

+ = =

. Do (1) ng

n n k= .Vy phng trnh ( ) 0f x = c v s nghim trn on [ ]0,1 .

Cu 66Cho hm s ( )f x lin tc trn tho mn iu kin:

( )( ) ( ) 1 xf f x f x = v ( )1000 999f = . Hy tnh ( )500f .Gii

Vi 1000x = , ta c: ( )( ) ( ) ( )1

1000 1000 1 999999

f f f f = = .

Xt hm s: ( ) ( ) 500g x f x=

f lin tc trn f lin tc trn [ ]999;1000 g lin tc trn [ ]999;1000 .

( ) ( )1

999 999 500 500 0999

g f= = <


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( ) ( )1000 1000 500 999 500 0g f= = >

Suy ra: ( ) ( ) ( ) ( )0 0999 . 1000 0 999;1000 : 0g g x g x< =

( ) ( )0 0999;1000 : 500 x f x = .

Thay 0x x= ta c ( )( ) ( ) ( )0 01

1 500500

f f x f x f = = .

Cu 67Cho hm s :f tho mn iu kin:

( ) ( ) ( )2 x, y

3

f x f y f xyx y

= + + (1) . Hy xc nh gi tr c th

c ca ( )2011f .

GiiCho 0x y= = thay vo (1) ta c:

( ) ( )( ) ( )

( )

( )

22

0 20 02 0 0 6 0

3 0 3

ff ff f

f

= = =

=

+ Xt ( )0 2f = . Khi :( ) ( ) ( )

( )0 0 3

2 23 2

f x f f x f x x

= + = .

Thay vo (1) thy khng tho.+ Xt ( )0 3f = , khi ( ) 3 f x x= + . Thay vo (1) thy tho mn.

Vy ( )2011 2011 3 2014f = + = .

Cu 68Cho hm s :f tho mn iu kin

( ) ( )3 32 2 x, y f x y f y x+ = + .Chng minh rng f l hm hng.GiiVi mi a, b thuc , chng minh tn ti ,x y sao cho:

3 32 , y 2 x y a x b+ = + = .

R rng ( ) ( ) f a f b f

=

l hm hng.

Xt h phng trnh:

323 3

33

22 02

222

a x x y a y a x

x b y x b

y x b

+ = =

+ = + = + =

.

y l phng trnh a thc bc l ( bc 9) i vi x nn lun c nghimtrn . Suy ra h trn lun c nghim (x, y).Vy f l hm hng.


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Cu 69Tm gi tr ca k sao cho tn ti hm lin tc :f tho mn:

( )( ) 9 x f f x kx= .Gii- Trng hp: k = 0 th hm ( ) 0 f x x= tho mn yu cu bi ton.

- Trng hp: 0k + fn iu+ f l mt n nh. Tht vy! ,x y ,

( ) ( ) ( )( ) ( )( ) 9 9 9 9 f x f y f f x f f y kx ky x y x y= = = = = .V f lin tc v l n nh nn fn iu thc s

Nu f tng thc s.Khi :

( ) ( ) ( )( ) ( )( ) ( )( )x y f x f y f f x f f y f f x< < < tng thc s. Nu f gim thc s

( ) ( ) ( )( ) ( )( ) ( )( )x y f x f y f f x f f y f f x< > < gim thc s.

Vy ( )( ) f f x l hm tng thc, v th 9 y kx= cng l hm tng thc s.Do 0k > .

Ngc li vi k > 0, ta lun tm c hm ( ) 34 x f x k x= .

Cu 70Tn ti hay khng hm s :f sao cho vi mi x, y thuc ta c:

( ) ( ){ } ( )( )max , min , f xy f x y f y x= + .GiiThay 1x y= = ta c

( ) ( ){ } ( ){ } ( )1 max 1 ,1 min 1 ,1 1 1 0 1 f f f f = + = + = ( V l).Vy hm f khng tn ti.Cu 71

Tm ( ) ( )1 2 2

0

min 1f

K x f x dx

= + , y [ ]( ) ( )1

0

0,1 : 1 f C f x dx = =

.

Gii p dng bt ng thc Schwarz ta c:

( ) ( ) ( ) ( )221 1 1 1

2 2 2

220 0 0 0

11 1 1 .

1 41

dx f x dx x f x dx x f x dx K

xx

= = + + =

+ +

Suy ra: ( ) ( )1

2 2

0

4min 1

fK x f x dx

= + .


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Cu72 Gi s rng f v g l cc hm kh vi trn [ ]a;b ; trong

( ) ( )g x 0 , g x 0 . Chng minh rng tn ti ( )c a;b sao cho:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

f a f b f c g cdet det

g a g b f c g c

g b g a g c

=

.

Gii

Xt hai hm s: ( )( )

( )( )

( )

f x 1h x , k x

g x g x= = kh vi trn [ ]a;b .

p dng nh l Cauchy ta c:

( ) ( ) ( )( ) ( )

( )( )

h b h a h cc a;b : k b k a k c =

( )

( )

( )

( )

( )

( ) ( )

( ) ( ) ( ) ( )

( )( )( )

( )( )

2

2

f c g c f c g cf b f ag cg b g a

c a;b :1 1 g c

g b g a g c

=

Cu 73 Chng minh rng: ( )f x arctan x= tho mn phng trnh:

( )

( )

( ) ( )

( )

( ) ( )( )

( )

( )

n n 1 n 22

1 x f x 2 n 1 f x n 2 n 1 f x 0

+ + + = v

i x

vn 2 .Gii

( )f x arctan x=

( ) ( ) ( )221

f x 1 x f x 11 x

= + =+

(1)

Ly o hm hai v ca (1) suy ra: ( ) ( ) ( )21 x f x 2xf x 0 + + = .Bng quy np ta chng minh c:

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )n n 1 n 221 x f x 2 n 1 xf x n 2 n 1 f x 0 + + + =

( )x , n 2

+ Mnh ng trong trng hp n = 2.+ Gi s mnh ng n n k= tc l: ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )k k 1 k 221 x f x 2 k 1 xf x k 2 k 1 f x 0 + + + = (*)Ly o hm hm hai v ca (*) ta c


33/35



( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( )

( ) ( )( )

( )

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

k k 1 k 12

k k 1

k 1 k k 12

2xf x 1 x f x 2 k 1 f x

2 k 1 xf x k 2 k 1 f x 0

1 x f x 2kxf x k 1 kf x 0

+

+

+ + +

+ + =

+ + + =

Cu 74 Cho f l hm kh vi n cp n trn ( )0;+ . Chng minh rng vi

x 0> ,

( ) ( )( )n

nn n 1

n 1

1 1 1f 1 x f

x x x

+

=

Gii+ Mnh ng trong trng hp n 1= .

+ Gi s mnh ng trong trng hp n k , tc l:( ) ( )

( )kkk k 1

k 1

1 1 1f 1 x f

x x x

+

=

+ Ta s chng minh mnh trn ng vi n k 1= + .Tht vy!

( )( )

( )

( )

( )( )

kk 1 k

k 1 k k 1k k k 1 k 21 1 1 11 x f 1 x f 1 kx f x f x x x x

+

+ +

= =

= ( )

( )

( )( )k k

k 1 k 1k 1 k 21 11 k x f 1 x f x x

+ +

( ) ( )( )k

k 1k k 2

k 1

k 1 1f 1 x f

x x x

+

=

.

Li c: ( )( )

( )( )k k 1

k 1 k 1k 2 k 21 11 x f 1 x f x x

=

Theo gi thit quy np vi trng hp n k 1= ta c:( ) ( )

( )k 1

k 1k k 2

k

1 1 1f 1 x f

x x x

=

.

T suy ra( )( )

( )

k 1k 1 k 1k

k 2

1 1 11 x f f

x x x

+

+ +

+

=

.

Vy bi ton c chng minh xong


34/35



Cu 75 Cho f kh vi trn ( )a;b sao cho vi ( )x a;b ta c:

( ) ( )( )f x g f x = , trong g ( )C a;b . Chng minh f C ( )a;b .GiiTa c: ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( )f x g f x f x g f x f x g f x g f x = = =

( ) ( )( ) ( )( )( ) ( )( )( ) ( )( )2 2

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

Do f , f u lin tc trn ( )a;b .

Chng minh bng quy np ta c ( ) ( )nf n 3 u l tng cc o hm( ) ( )kg f vi k 0;n 1= . T suy ra iu phi chng minh.

Cu 76 Cho [ ]f : ; 1;12 2

l mt hm kh vi c o hm lin tc v

khng m. Chng minh tn ti 0x ;2 2

sao cho

( )( ) ( )( )2 2

0 0f x f x 1+ .

Gii

Xt hm s:

( )

g : ; ;2 2 2 2

x arctan f x

g l hm lin tc trn ;2 2

. Nu ( )f x 1 th g kh vi ti mi x v

( )( )

( )( )

f xg x

1 f f x

=

.

Nu tn ti 0x ;2 2

sao cho

( )

( )

0

0

f x 1

f x 1

=

= th0x l cc tra phng

ca hm f nn theo nh l Fermat ta suy ra c ( )0f x 0 = . V th ta c:

( )( ) ( )( )2 2

0 0f x f x 1+ = .

Nu ( )f x 1 x ;2 2

th p dng nh l Lagrange cho hm g trn

on ;2 2

:


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( )

( )( )

00 2

0

f xx ; : g g

2 2 2 2 2 21 f x

=

.

D thy:( )

( )( )

0

2

0

f x0

1 f x

.

Vy ta chng minh c ( )( ) ( )( )2 2

0 0f x f x 1+ .

Cu 77Cho f kh vi trn [ ]a;b v tho mn:

a) ( ) ( )f a f b 0= = b) ( ) ( ) ( ) ( )f a f a 0 , f b f b 0+ = > = > .

Chng minh rng tn ti ( )c a;b sao cho ( )f c 0= v ( )f c 0 .GiiT gi thit suy ra f bng 0 ti t nht mt im trong khong ( )a;b .

t ( ) ( ){ }c inf x a;b : f x 0= = , ta c ( )f c 0= .

V ( )f a 0 > nn ( ) ( )f x 0 x a;c> . Hn na ( )f c tn ti nn

( )( ) ( ) ( )

h 0 h 0

f c h f c f c hf c lim lim 0

h h + +

= = .

Cu 78

Cho ( )f x l hm s c o hm ti im 0x 2011= v n . Chng minh

rng: ( ) ( )n

1 2011nlimn f f 2011 f 2011

n+

=

.

GiiV f c o hm ti im 0x 2011= nn theo nh ngha ta c:

( ) ( )( )0 0x 0

f 2011 x f xlim f x

x +

=

Xt ring: Nu ly1

x

n

= , ta c x 0 khi n .

Ta c:

( )( )

( )n n

1f 2011 f 2011

1 2011n nlimn f f 2011 lim f 20111nn

+ + = =

.

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