ÔN THI ĐẠI HỌC - TUYỆT CHIÊU HÀM SỐ 12 -2009

8/14/2019 N THI I HC - TUYT CHIU HM S 12 -2009

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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s

5

TM TT L THUYT1. nh ngha :Gi s K l mt khong , mt on hoc mt na khong . Hm s f xc nh trn Kc gi l

ng bin trn K nu vi mi ( ) ( )1 2 1 2 1 2, ,x x K x x f x f x < 2. iu kin cn hm s n iu :Gi shm s f c o hm trn khong I

Nu hm s f ng bin trn khong I th ( )' 0f x vi mi x I Nu hm s f nghch bin trn khong I th ( )' 0f x vi mi x I 3. iu kin hm s n iu :nh l 1 : nh l v gi tr trung bnh ca php vi phn (nh l Lagrange):

Nu hm s f lin tc trn ;a b v c o hm trn khong ( );a b th tn ti t nht mt im ( );c a b

sao cho ( ) ( ) ( ) ( )'f b f a f c b a =

nh l 2 :Gi s I l mt khong hoc na khong hoc mt on , f l hm s lin tc trn I v c o hm timi im trong ca I ( tc l im thuc I nhng khng phi u mt ca I ) .Khi :

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

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

Nu hm s f lin tc trn ;a b v c o hm ( )' 0f x < trn khong ( );a b th hm s f nghchbin trn ;a b

TNH N IU CA HM S


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6

CC BI TON C BN

V d 1:

Xt chiu bin thin ca cc hm s :

Gii :

( )3 21

) 3 8 23a f x x x x = + Hm s cho xc nh trn .

Ta c ( ) 2' 6 8 f x x x = +

( )' 0 2, 4 f x x x = = = Chiu bin thin ca hm s c nu trong bng sau :x 2 4 +

( )'f x + 0 0 + ( )f x +

Vy hm s ng bin trn mi khong ( );2 v ( )4;+ , nghch bin trn khong ( )2;4

( )2 2

)1

x xb f x

x

=

Hm s cho xc nh trn tp hp { }\ 1 .

Ta c ( )

( )

( )

( )

22

2 2

1 12 2' 0, 1

1 1

xx x f x x

x x

+ += = >

Chiu bin thin ca hm s c nu trong bng sau :x 1 +

( )'f x + + + +

( )f x

( ) 3 21

) 3 8 2

3

a f x x x x = +

( )2 2

)1

x xb f x

x

=

( ) 3 2) 3 3 2c f x x x x = + + +

( ) 3 21 1

) 2 23 2

d f x x x x = +


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7

Vy hm s ng bin trn mi khong ( );1 v ( )1;+

( ) 3 2) 3 3 2c f x x x x = + + + Hm s cho xc nh trn .

Ta c ( ) ( )2

2' 3 6 3 3 1 f x x x x = = + = +

( )' 0 1 f x x = = v ( )' 0f x > vi mi 1x

V hm s ng bin trn mi na khong ( ; 1 v )1; + nn hm s ng bin trn .Hoc ta c th dng bng bin thin ca hm s :x 1 +

( )'f x + 0 + ( )f x +

1

V hm s ng bin trn mi na khong ( ; 1 v )1; + nn hm s ng bin trn .

( ) 3 21 1

) 2 23 2

d f x x x x = + Tng t bi )a

V d 2:

Gii :

( ) 3 2) 2 3 1a f x x x = + +

Hm s cho xc nh trn .Ta c ( ) 2' 6 6 f x x x = +

( ) ( ) ( ) ( )' 0, ; 1 , 0; f x x f x > + ng bin trn mi khong ( ); 1 v ( )0;+ .

( ) ( ) ( )' 0, 1;0 f x x f x < nghch bin trn khong ( )1;0 .

Ngoi ra : Hc sinh c th gii ( )' 0f x = , tm ra hai nghim 1, 0x x= = , k bng bin thin ri ktlun.

Xt chiu bin thin ca cc hm s :

( ) 3 2) 2 3 1a f x x x = + +

( ) 4 2) 2 5b f x x x =

( ) 3 24 2

) 6 93 3

c f x x x x = +

( ) 2) 2d f x x x =


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8

( ) 4 2) 2 5b f x x x = Hm s cho xc nh trn .

Ta c ( ) 3' 4 4 f x x x =

( ) ( ) ( ) ( )' 0, 1;0 , 1; f x x f x > + ng bin trn mi khong ( )1;0 v ( )1; + .

( ) ( ) ( ) ( )' 0, ; 1 , 0;1 f x x f x < nghch bin trn mi khong ( ); 1 v ( )0;1 .

Ngoi ra : Hc sinh c th gii ( )' 0f x = , tm ra hai nghim 1, 0, 1x x x= = = , k bng bin thin rikt lun.

( ) 3 24 2

) 6 93 3

c f x x x x = +

Hm s cho xc nh trn .

Ta c ( ) ( )2

2' 4 12 9 2 3 f x x x x = + =

( )3

' 0 2 f x x = = v ( )' 0f x < vi mi3

2x

V hm s nghch bin trn mi na khong3

;2

v

3;

2

+

nn hm s nghch bin trn .

( ) 2) 2d f x x x =

Hm s cho xc nh trn 0;2 .

Ta c ( ) ( )2

1' , 0;2

2

x f x x

x x

=

( ) ( ) ( )' 0, 0;1 f x x f x >

ng bin trn khong ( )0;1 ( ) ( ) ( )' 0, 1;2 f x x f x < nghch bin trn khong ( )1;2

Hoc c th trnh by :

( ) ( ) ( )' 0, 0;1 f x x f x > ng bin trn on 0;1

( ) ( ) ( )' 0, 1;2 f x x f x < nghch bin trn on 1;2

V d 3:

Gii :

D thy hm s cho lin tc trn on 0;2 v c o hm ( ) 2' 04x

f xx

= + >

l hm s ng bin trn

0;2

v ( ) ( )0 , 0;

2 f x f x

>

hay sin tan 2 , 0;

2x x x x

+ >

.

NG DNG O HM TRONG CC BI TON I S

V d 1:

Gii :

t2sin ; 0 1t x t= .

Khi phng trnh ( ) 5 581

* 81 (1 ) , 0;1256

t t t + =

Xt hm s5 5( ) 81 (1 ) f t t t = + lin tc trn on 0;1 , ta c:

4 4'( ) 5[81 (1 ) ],t 0;1 f t t t =

4 481 (1 ) 1

'( ) 0

40;1

t t f t t

t

= = =

Lp bng bin thin v t bng bin thin ta c:1 81

( ) ( )4 256

f t f =

Vy phng trnh c nghim21 1 1

sin cos2 ( )4 4 2 6

t x x x k k Z

= = = = + .

Chng minh rng : sin tan 2 , 0;2

x x x x

+ >

.

Gii phng trnh : ( )10 1081

81sin cos *256

x x+ =


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11

V d 2:

Gii :

2 21. 3 (2 9 3) (4 2)( 1 1) 0 (1)x x x x x + + + + + + + = Phng trnh (1) ( )

2 23 (2 ( 3 ) 3) (2 1)(2 (2 1) 3) (2)x x x x + + = + + + + t 3 , 2 1, , 0u x v x u v = = + >

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

Xt hm s 4 2( ) 2 3 , 0 f t t t t t = + + >

Ta c ( )3

4 2

2 3'( ) 2 0, 0

3

t t f t t f t

t t

+= + > >

+ng bin trn khong ( )0;+ .

Khi phng trnh (3) 1( ) ( ) 3 2 15

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

Vy1

5x = l nghim duy nht ca phng trnh.

Ch :

Nu hm s ( )y f x= lun n iu nghim ngoc ( hoc lun ng bin hoc lun nghch bin ) th

s nghim ca phng trnh : ( ) f x k = s khng nhiu hn mt v ( ) ( ) f x f y = khi v ch khi

x y=

.

2tan2. os =2 , - ;2 2

xe c x x

+

Xt hm s :

2tan( ) osx f x e c x = + lin tc trn khong - ;2 2

x

. Ta c

Gii phng trnh :

2 21. 3 (2 9 3) (4 2)( 1 1) 0x x x x x + + + + + + + =

2tan2. osx=2 , - ;2 2

xe c x +

.3. 2003 2005 4006 2x x x+ = +

34. 3 1 log (1 2 )x x x= + + +


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12

23

2

3

tantan

2

1 2e os'( ) 2 tan . sin sin

cos os

xx c x f x x e x x

x c x

= =

V2

3tan2 2 os 0xe c x > >

Nn du ca '( )f x chnh l du ca sin x. T y ta c ( ) (0) 2 f x f = Vy phng trnh cho c nghim duy nht 0x = .

3. 2003 2005 4006 2x x x+ = +

Xt hm s : ( ) 2003 2005 4006 2x x f x x = +

Ta c: '( ) 2003 ln2003 2005 ln2005 4006x xf x = +

2 2''( ) 2003 ln 2003 2005 ln 2005 0 "( ) 0x x f x x f x = + > = v nghim

( )' 0f x = c nhiu nht l mt nghim . Do phng trnh ( ) 0f x = c nhiu nht l hai nghim

v ( ) ( )0 1 0f f= = nn phng trnh cho c hai nghim 0, 1x x= = Ch :

Nu hm s ( )y f x= lun n iu nghim ngoc ( hoc lun ng bin hoc lun nghch bin )

v hm s ( )y g x= lun n iu nghim ngoc ( hoc lun ng bin hoc lun nghch bin ) trn

D, th s nghim trn D ca phng trnh ( ) ( ) f x g x = khng nhiu hn mt.

Nu hm s ( )y f x= ) c o hm n cp n v phng trnh ( )( ) 0kf x = c m nghim, khi

phng trnh ( 1)( ) 0kf x = c nhiu nht l 1m + nghim

34. 3 1 log (1 2 )x x x= + + +

1

2x >

Phng trnh cho

( )3 3 33 1 2 log (1 2 ) 3 log 3 1 2 log (1 2 ) *x x xx x x x x + = + + + + = + + +

Xt hm s:3

( ) log , 0 f t t t t = + > ta c ( ) ( )1' 1 0, 0ln 3

f t t f t t

= + > > l hm ng bin

khong ( )0;+ nn phng trnh

( ) ( )* (3 ) (1 2 ) 3 2 1 3 2 1 0 * *x x x f f x x x = + = + = Xt hm s: 2( ) 3 2 1 '( ) 3 ln 3 2 "( ) 3 ln 3 0x x x f x x f x f x = = = >


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( ) 0f x = c nhiu nht l hai nghim, v ( )(0) 1 0f f= = nn phng trnh cho c hainghim 0, 1x x= = .

V d 3:

Gii :

iu kin 2 3 2 0 1 2x x x x +

t 2 3 2, 0u x x u = +

Phng trnh ( ) ( ) ( ) ( )

2

2

1

3 3

1 1

* log 2 2 log 2 .5 2, 0 * *5 5

u

u

u u u

+ + = + + =

Xt hm s : ( ) ( )2

3

1log 2 .5

5u f u u

= + +

lin tc trn na khong )0; + , ta c :

( )2' 1 1( ) 5 . ln 5.2 0, 0

( 2)ln 3 5u f u u u f u

u= + >

+ng bin trn na khong )0; + v

( )1 2 1f u= = l nghim phng trnh ( )* * .

Khi 2 2

3 5

23 2 1 3 1 0 3 5

2

x

x x x x x

=

+ = + = +=

tho iu kin.

V d 4:

Gii phng trnh : ( ) ( )23 1

2

3

1log 3 2 2 2 *

5

x x

x x

+ + + =

Gii h phng trnh :

1.2 3 4 4 (1)

2 3 4 4 (2)

x y

y x

2. ( )( )

3

32 12 2

x x yy y x

+ = + =

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =


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14

Gii :

1.2 3 4 4 (1)

2 3 4 4 (2)

x y

y x

iu kin:

34

23

42

x

x

.

Cch 1:

Tr (1) v (2) ta c:

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

3( ) 2 3 4 , ; 4

2 f t t t t

= +

, ta c:

/ 1 1 3( ) 0, ; 422 3 2 4

f x t t t

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

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

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

22

39 02 2 5 12 9 11

9 38 33 09

xxx x x

x x x

= + + = + = =

Vy h phng trnh c 2 nghim phn bit

113

9,3 11

9

xx

yy

= =

= =

.

Cch 2:

Tr (1) v (2) ta c:

( ) ( )2 3 2 3 4 4 0x y y x + + + = (2 3) (2 3) (4 ) (4 ) 02 3 2 3 4 4

x y y x

x y y x

+ + + =

+ + + +

2 1( ) 02 3 2 3 4 4

x y x y x y y x

+ = = + + + +

.

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

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

22

39 02 2 5 12 9 11

9 38 33 09

xxx x x

x x x

= + + = + = =


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15


113

9,3 11

9

xx

yy

= =

= =

.

2.

( )

( )

3

3

2 1

2 2

x x y

y y x

+ = + =

Cch 1 :

Xt hm s 3 / 2( ) 2 ( ) 3 2 0, f t t t f t t t = + = + > .

H phng trnh tr thnh( ) (1)

( ) (2)

f x y

f y x

= =

.

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

Suy ra x y= , th vo h ta c ( )3 2 20 1 0 0 1 0.x x x x x v x + = + = = + > Vy h c nghim duy nht

0

0

x

y

= =

.

Cch 2:

Tr (1) v (2) ta c:3 3 2 23 3 0 ( )( 3) 0x y x y x y x y xy + = + + + =

2 23( ) 3 0

2 4

y yx y x x y

+ + + = =

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

Vy h phng trnh c nghim duy nht0

0

x

y

= =

.

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =

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

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

Xt hm s3

( ) 3 f t t t = lin tc trn on [ 1;1] , ta c

( )2'( ) 3( 1) 0 [ 1;1] f t t t f t = nghch bin trn on [ 1;1]

Do : (*) x y = thay vo (2) ta c nghim ca h l:6

1

2x y= = .

V d 5:


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Gii :

1.2

1 1(1)

2 1 0 (2)

x yx y

x xy

= =

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

1(1) ( ) 1 0 1

.

y x

x yxy y

x

= + = =

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

1

yx

= phng trnh (2)v nghim.

Vy h phng trnh c 2 nghim phn bit1 1;

1 1

x x

y y

= =

= = .

Bnh lun:

Cch gii sau y sai:2

1 1(1)

2 1 0 (2)

x yx y

x xy

= =

.

iu kin: 0, 0x y .

Xt hm s /2

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

t t= = + > .

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

Sai do hm s ( )f t n iu trn 2 khong ri nhau (c th ( ) ( )1 1 0f f = = ).

2.3

1 1(1)

2 1 (2)

x yx y

y x

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

Gii h phng trnh :

1.2

1 1(1)

2 1 0 (2)

x yx y

x xy

= =

2.3

1 1(1)

2 1 (2)

x yx y

y x

= = +


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1 1(1) 0 ( ) 1 0 .

x yx y x y x y y

xy xy x

+ = + = = =

x y= phng trnh (2)1 5

1 .2

x x

= =

1

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

Xt hm s 4 / 33

1( ) 2 ( ) 4 1 0 .

4 f x x x f x x x

= + + = + = =

4

3 3

1 32 0, lim lim ( ) 0, 2 0

4 4 4 x x f f x x x x

+

= > = = + > + + =

v nghim.

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

1 1(1) 0 ( ) 1 0 .x yx y x y x y y xy xy x

+ = + = = =

x y= phng trnh (2)1 5

1 .2

x x

= =

1

yx

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

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

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

Suy ra phng trnh (2)v nghim.


1 5 1 51

2 21 1 5 1 5

2 2

x xx

yy y

+ = = = = + = =

.

V d 6:

Gii h phng trnh:

1.2 1

2 1

2 2 3 1( , )

2 2 3 1

y

x

x x xx y R

y y y

+ + = +

+ + = +

2.

2 1 2 2 1

3 2

(1 4 )5 1 2 (1)

4 1 ln( 2 ) 0 (2)

x y x y x y

y x y x

+ + + = +

+ + + + =


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Gii :

1.2 1

2 1

2 2 3 1( , )

2 2 3 1

y

x

x x xx y R

y y y

+ + = +

+ + = +

t 1, 1u x v y = =

( )I vit li2

2

1 3( )

1 3

v

u

u uII

v v

+ + = + + =

Xt hm s : ( ) 2 1 f x x x = + + lin tc x , ta c

( ) ( )2

2 2 2

11 0,

1 1 1

x xx x x f x x f x

x x x

++ += + = >

+ + + ng bin x .

Nu ( ) ( ) 3 3v uu u f u f v v u > > > > v l

Tng t nu v u> cng dn n v l

Do h ( )2 21 3 1 3 ( 1 ) (1)

IIu uu u u u

u v u v

+ + = = +

= =

t: ( ) 23 ( 1 )ug u u u = + lin tc u R , ta c

2 2

2 2

1'( ) 3 ln 3( 1 ) 3 1 3 1 ln 3 0,

1 1

u u uug u u u u u u R

u u

= + + = + > + +

Do ( )g u ng bin u R v ( )0 1 0g u= = l nghim duy nht ca ( )1 .Nn ( )II 0u v = = . Vy ( ) 1I x y = =

2.

2 1 2 2 1

3 2

(1 4 )5 1 2 (1)

4 1 ln( 2 ) 0 (2)

x y x y x y

y x y x

+ + + = +

+ + + + =

t 2t x y= . Khi phng trnh (1) tr thnh: ( )1 4

5[( ) ( ) ] 1 2.2 *5 5

t t t+ = +

Xt ( ) 1 45[( ) ( ) ]5 5

t tf t = + , ( ) 1 2.2tg t = +

D thy : ( )1 4

5[( ) ( ) ]5 5

t tf t = + l hm nghch bin v ( ) 1 2.2tg t = + l hm ng bin

v ( ) ( )1 1 5 1 f g t = = = l mt nghim ca ( )* . Do ( )* c nghim duy nht 1t = .

1 2 1 2 1t x y x y = = = + khi : ( )3 2(2) 2 3 ln( 1) 0 * *y y y y + + + + + =


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19

Xt hm s3 2

( ) 2 3 ln( 1) f y y y y y = + + + + + , ta c:2

2 2

2 2

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

1 1

y y y f y y y f y

y y y y

+ + += + + = + >

+ + + +l hm ng bin

v ( 1) 0f =

nn ( )* * c nghim duy nht 1y=

Vy nghim ca h l:0

1

x

y

=

=

.

V d 7:

Gii :

t: ( ) ( )2

,1

t tf t e g t t

= =

lin tc trn khong ( )1, + , ta c

( ) ( )' 0, 1t f t e t f t = > > ng bin trn khong ( )1, +

( )/ 32 2

1( ) 0, 1

( 1)

g t t g t

t

= < >

nghch bin trn khong ( )1, + .

H phng trnh ( )( ) ( )( ) ( )

( ) ( ) ( ) ( )2

2

200720071 12007

20071

x

y

ye

f x g y yf x g y f y g x

f y g x xe

x

= + = + = +

+ = =

Nu ( ) ( ) ( ) ( )x y f x f y g y g x y x > > < > v l.Tng t y x> cng v l .

Khi ( ) ( )2

2

2

20072007 01 1 2

12007

1

x

x

y

ye x

eyxx

x yex

= + =

= =

Xt hm s: ( )2

20071

x xh x ex

= +

lin tc trn khong ( )1; + , ta c

Chng minh rng h phng trnh ( )2

2

2007

1 12007

1

x

y

ye

yx

ex

=

=

c ng 2 nghim tha mn iu kin

1, 1x y> >


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20

( )( )

( ) ( ) ( )( )

3 5

2 22 2

3 5

2 22 2

1 3 3' 1 , '' 1 .2 0

21 1

x x x x xh x e e x h x e x x e

x x

= = = + = + >

v ( ) ( )1

lim , limxx

h x h x + +

= + = +

Vy ( )h x lin tc v c th l ng cong lm trn ( )1;+ .Do chng minh ( )2 c 2 nghim ln hn 1 ta ch cn chng minh tn ti 0 1x > m ( )0 0h x < .

Chn ( ) ( )202

2 : 2 2007 0 03

x h e h x = = + < = c ng hai nghim 1x >

Vy h phng trnh ( )1 c ng 2 nghim tha mn iu kin 1, 1x y> > .

V d 8:

Gii :

1.

2

2

2

2

12

1

21

xy

xy

zy

zxz

=

=

=

Gi s x y z> >

Xt hm s : ( ) 22

1

tf t

t=

,xc nh trn { }\ 1D = .Ta c

( ) ( )2

2 2

2( 1)0,

(1 )

t f t x D f t

t

+= >

lun ng bin trn D.

Gii h phng trnh sau:

1.

2

2

2

2

12

12

1

xy

xy

zy

zx

z

=

=

=

2.

3 2

3 2

3 2

9 27 27 0

9 27 27 09 27 27 0

y x x

z y yx z z

+ =

+ = + =


17/42


21

Do : ( ) ( ) ( )x y z f x f y f z y z x > > > > > > . Mu thun, do iu gi s sai .Tng t x y z< < khng tho .Vy x y z= =

H cho c nghim : ( ) ( ); ; 0;0;0x y z =

2.

3 2

3 2

3 2

9 27 27 0

9 27 27 0

9 27 27 0

y x x

z y y

x z z

+ = + =

+ =

3 2 3 2

3 2 3 2

3 2 3 2

9 27 27 0 9 27 27

9 27 27 0 9 27 27

9 27 27 0 9 27 27

y x x y x x

z y y z y y

x z z x z z

+ = = +

+ = = + + = = +

Xt hm s c trng : 2( ) 9 27 27 '( ) 18 27 f t t t f t t = + =

( )

3'( ) 0,3 2'( ) 0 18 27 032

' 0,2

f t t f t t t

f t t

> >= = =

< + >

>

Vy , ,x y z thuc min ng bin, suy ra h phng trnh

( )

( )

( )

f x y

f y z

f z x

=

= =

l h hon v vng quanh.

Khng mt tnh tng qut gi s3 3( ) ( )x y f x f y y z y z

3 3( ) ( ) f y f z z x z x

x y z x x y z = =

Thay vo h ta c: 3 29 27 27 0 3x x x x + = = .Suy ra: 3x y z= = = V d 9:

Gii h phng trnh :

1.

3 2

3 2

3 3 ln( 1)

3 3 ln( 1)

x x x x y

y y y y z

+ + + = + + + =


18/42


22

Gii :3 2

3 2

3 2

3 3 ln( 1)1. 3 3 ln( 1)

3 3 ln( 1)

x x x x y y y y y z

z z z z x

+ + + =

+ + + =

+ + + =

H phng trnh c dng :

( )

( )

( )

f x y

f y z

f z x

=

=

=

.

Ta gi s ( ); ;x y z l nghim ca h. Xt hm s3 2

( ) 3 3 ln( 1), f t t t t t t R= + + + .

Ta c: ( )22

2 1'( ) 3 3 0,

2 1

t f t t t f t

t t

= + + >

+ l hm ng bin t R .

Gi s: { }max ; ;x x y z = th ( ) ( ) ( ) ( )y f x f y z z f y f z x = = = =

Vy x y z= = . V phng trnh 3 22 3 ln( 1) 0x x x x + + + =

Xt hm s ( ) 3 22 3 ln( 1),g x x x x x x R= + + + , hm s ( )g x ng bin trn R v

( )1 0g = , do phng trnh ( ) 0g x = c nghim duy nht 1x = .Do h cho c nghim l 1x y z= = = .

23

23

23

2 6 log (6 )

2. 2 6 log (6 )

2 6 log (6 )

x x y x

y y z y

z z x z

+ =

+ =

+ =


19/42


23

H cho

32

32

32

log (6 )2 6 ( ) ( )

log (6 ) ( ) ( )

2 6 ( ) ( )log (6 )

2 6

xy

x x f y g x y

z f z g y

y y f x g z zx

z z

= + =

= = +

= =

+

Xt hm s3

2( ) log (6 ) ; ( ) , ( ;6)

2 6

t f t t g t t

t t

= = +

Ta c( )

( )1

'( ) 0, ( ;6)6 ln 3

f t t f t t

= <

nghch bin trn khong ( ;6) v

( )( )3

2

6'( ) 0, ( ;6)

2 6

tg t t g t

t t

= >

+

ng bin trn khong ( ;6) .

Ta gi s ( ); ;x y z l nghim ca h th x y z= = thay vo h ta c:

32

log (6 ) 32 6

xx x

x x

= = +

Vy nghim ca h cho l 3x y z= = = .

Ch :H HON V VNG QUANH:

nh ngha: L h c dng:

1 2

2 3

1

( ) ( )

( ) ( )

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

( ) ( )n

f x g x

f x g x

f x g x

=

= =

(I)

nh l 1:Nu ,f gl cc hm cng tng hoc cng gim trn A v 1 2( , ,..., )nx x x l nghim ca h

trn A th1 2

...n

x x x= = =

nh l 2:Nu ,f gkhc tnh n iu trn A v 1 2( , ,..., )nx x x l nghim ca h trn A th

1 2...

nx x x= = = nu n l v 1 3 1

2 4

...

...n

n

x x x

x x x

= = =

= = =nu n chn

V d 10:

Gii h phng trnh :

1.

sin sin 3 3 (1)

(2)

x y x y

x y

=

+ =


20/42


21/42


25

t cos ; sinu x v y = = , ta c h:( )

( )2 3

2 3

log (1 3 ) log ( ) 2 1

log (1 3 ) log ( ) 2 2

u v

v u

+ = +

+ = +

tr v theo v ta c

( )3 3 3 3

log (1 3 ) log log (1 3 ) log ( ) ( ) *u u v v f u f v + + = + + =

Xt hm s 3 3( ) log (1 3 ) log f t t t = + + , d thy ( )f t l hm ng bin nn ( )* u v = .

Thay vo ( )1 ta c :3 3

1 3 1log (1 3 ) log 2 9

6

uu u u

u

++ = = =

Vy h cho

21sin

6 21

cos 26

y ky

y k

x x m

= +=

= + = = +

, trong 1

sin cos6

= = .

V d 11:

Gii :

Xt hm s ( ) 4 2 1 f x x x = + lin tc trn na khong )0; + , ta c

( )( )

324

1 1' 02

1

xf xxx

=


22/42


26

iu kin: 3 1x .

Phng trnh3 3 4 1 1

(1)4 3 3 1 1

x xm

x x

+ + + =

+ + +

Nhn thy rng: ( ) ( )

2 22 2 3 1

3 1 4 12 2

x x

x x

+ + + = + =

Nn tn ti gc 0; , tan ; 0;12 2

t t

=

sao cho:

2

23 2sin 2

1

tx

t+ = =

+v

2

2

11 2 cos 2

1

tx

t

= =

+

( ) ( )2

2

3 3 4 1 1 7 12 9, 2

5 16 74 3 3 1 1

x x t t m m f t

t tx x

+ + + + += = =

+ ++ + +

Xt hm s:2

27 12 9( )5 16 7

t tf tt t

+ += + +

lin tc trn on 0;1t . Ta c

( )( )

2

22

52 8 60'( ) 0, 0;1

5 16 7

t t f t t f t

t t

= <

+ +nghch bin trn on [ ]0;1 v

9 7(0) ; (1)

7 9f f= =

Suy ra phng trnh ( )1 c nghim khi phng trnh ( )2 c nghim trn on 0;1t khi v ch khi:7 9

9 7m

CC BI TON BT PHNG TRNH

V d 1:

Gii:

1. 5 1 3 4x x + +

iu kin :1

5x

Gii cc bt phng trnh sau :

1. 5 1 3 4x x + +

52. 3 3 2 2 6

2 1x x

x +


23/42


27

Xt hm s ( ) 5 1 3 f x x x = + + lin tc trn na khong1;

5

+

, ta c

( )5 1 1

'( ) 0 ,52 5 1 2 1

f x x f x x x

= + > >

l hm s ng bin trn na khong

1;

5 +

v (1) 4f = , khi bt phng trnh cho ( ) (1) 1. f x f x

Vy bt phng trnh cho c nghim l 1x .

52. 3 3 2 2 6

2 1x x

x +

V d 2:

Gii:

iu kin:1 3

2 2x<

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

x x f x g x x

+ +

Xt hm s5

( ) 3 3 22 1

f x x x

= +

lin tc trn na khong1 3;

2 2

, ta c

3

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

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

x x

= <

l hm nghch bin trn na on1 3;

2 2

Hm s ( ) 2 6g x x= + l hm ng bin trn v (1) (1) 8f g= =

Nu 1 ( ) (1) 8 (1) ( ) (*)x f x f g g x > < = = < ng

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

Vy nghim ca bt phng trnh cho l:3

12

x .


52. 3 3 2 2 6

2 1x x

x +


24/42


28

V d 3:

Gii:

iu kin:1

2x .

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

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

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

( )1 1 2 6

'( ) ( )( 2 1 3) 0, 52 2 2 6 2 1

x x f x x x f x

x x x

+ + += + + > >

+ + ng

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

Vy nghim ca bt phng trnh cho l:1

72

x .

V d 3:

Gii :

iu kin:

3 22 3 6 16 0

2 4.4 0

x x xx

x

+ + +

.

Bt phng trnh cho ( )3 22 3 6 16 4 2 3 ( ) 2 3 *x x x x f x + + + < <

Xt hm s3 2

( ) 2 3 6 16 4 f x x x x x = + + + lin tc trn on 2;4 , ta c:

( ) ( )2

3 2

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

2 42 3 6 16

x x f x x f x

xx x x

+ += + >

+ + +ng bin trn na

khong ( )2;4 v (1) 2 3f = , do ( )* ( ) (1) 1 f x f x < < .


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


3 24. 2 3 6 16 2 3 4x x x x + + + < +


25/42


29

Vy nghim ca bt phng trnh cho l: 2 1x < .

CC BI TON LIN QUAN THAM SV d 1:

Gii:

1.Hm s cho xc nh trn .

Ta c ( ) 2' 2 4 f x x ax = + +

Cch 1 :

Hm s ( )f x ng bin trn khi v ch khi

( ) 2 2' 0, 2 4 0, 0 4 0 2 2 2 f x x x ax x a a hay a + + Cch 2 :

2 4a = Nu 2 4 0 2 2a hay a < < . Hm s ( )f x ng bintrn mi na khong ( ); 2 2;v + nn hm s ( )f x ng bin trn Nu 2a = . Hm s ( )f x ng bin trn Nu 2a < hoc 2a > th ( )' 0f x = c hai nghim phn bit 1 2,x x . Gi s 1 2x x< . Khi hm snghch bin trn khong

( )1 2;x x ,ng bin trn mi khong

( )1;x v

( )2;x + . Do 2a < hoc

2a > khng tho mn yu cu bi ton .

Vy hm s ( )f x ng bin trn khi v ch khi 2 2a Ch : li gii cch 1 thiu t nhin, khng trong sng .2.Hm s cho xc nh trn .

Ta c : ( ) ( ) ( ) ( )2 2' 1 2 1 3 f x a x a x g x = + + + =

Vi gi tr no ca ahm s sau ng bin trn .

( ) 3 21

1. 4 33

f x x ax x = + + +

( ) ( ) ( )2 3 21

2. 1 1 3 53

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


26/42


30

Hm s ( )f x ng bin trn khi v ch khi ( ) ( )' 0, 1 f x x Xt 2 1 0 1a a = =

( ) ( )3

1 ' 4 3 , ' 0 14

a f x x f x x a + = = + = khng tho yu cu bi ton.

( )1 ' 3 0 1a f x x a + = = > =

tho mn yu cu bi ton.

Xt 2 1 0 1a a ( )

( ) ( ) ( )

2

2 22

1 0 1 1 1 11 1 2

1 22 2 0' 1 3 1 0g

a a a a aa a

a aa aa a

> < > < > <

+ + = +

Kt hp cc trng hp , vi 1 2a a th th ca hm s ng bin trn .

V d 2:

Gii :1.

Hm s cho xc nh trn { }\ 1D = .

Ta c ( )( ) ( )

( )

( )

( )( ) ( ) ( )

2

2

2 2

1 2 1 1' , 1 2 1 1, 1

1 1

m x m x g x f x g x m x m x x

x x

+ += = = + +

+ +

Du ca ( )'f x l du ca ( )g x .

Hm s ( )f x ng bin trn mi khong ( ) ( ); 1 1;v + khi v ch khi ( ) ( )0, 1 1g x x Xt ( ) ( )1 0 1 1 0, 1 1m m g x x m a = = = > = tho mn yu cu bi ton .Xt 1 0 1m m

( )( ) ( ) ( ) ( )

( )21 0 1 1

1 1 21 21 2 0' 1 1 0

g

m m mm b

mm mm m

> > > <

=

T ( ) ( )a v b suy ra 1 2m th tho mn yu cu bi ton .2.Hm s cho xc nh trn { }\D m= .

1. Vi gi tr no ca mhm s ( ) ( )21 2 1

1

m x xf x

x

+ +=

+ng bin mi khong xc nh .

2. Vi gi tr no ca mhm s ( ) 4mxf xx m

+=

+nghch bin khong ( );1 .


27/42


31

Ta c ( )( )

2

2

4'

mf x

x m

=

+

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

( )' 0, ;1

;1

f x x

m

<

( )

2 4 0 2 2 2 22 1

1 1;1

m m mm

m mm

< < < < .

Xt hm s ( )2

6 4g x x x = lin tc trn khong ( )1;+ , ta c( ) ( )' 12 4 0, 1g x x x g x = > > ng bin trn khong ( )1;+ v

( ) ( ) ( )21 1

lim lim 6 4 2, limxx x

g x x x g x + + +

= = = +

x 1 +( )'g x +

( )g x +

2 Da vo bng bin thin suy ra 2 2m m 2.Hm s cho xc nh trn .

Ta c : 2' 3 2 3y mx x = +

Hm s cho ng bin trn khong ( )3;0 khi v ch khi ( )' 0, 3;0y x

Hay ( ) ( )2 22 3

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

xmx x x m x

x

+ +

Tm iu kin ca tham s msao cho hm s :

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


28/42


32

Xt hm s ( ) 22 3

3

xg x

x

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

( ) ( ) ( )2

4

6 18' 0, 3;0

9

x xg x x g x

x

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

( ) ( )3 04

lim , lim9x xg x g x + = = x 3 0

( )'g x ( )g x 4

9

Da vo bng bin thin suy ra4

9m

V d 4 :

Gii :


Ta c : ( ) 2' 3 6 1 f x x x m = + + +

Cch 1 :

Hm s cho nghch bin trn khong ( )1;1 khi v ch khi ( ) ( )' 0, 1;1 f x x hay

( ) ( )( )

( ) ( )21;1

3 6 1 , 1;1 min 1x

m x x x m g x

+ + . Xt hm s( ) ( ) ( ) ( ) ( ) ( )23 6 1 , 1;1 ' 6 6 0, 1;1g x x x x g x x x g x = + + = < nghch bin trn

khong ( )1;1 v ( ) ( )1 1

lim 2, lim 10x x

g x g x +

= =

x 1 1( )'g x

( )g x 2 10


1. ( )3 23 1 4y x x m x m = + + + + nghch bin trn khong ( )1;1 ?.2. ( ) ( )3 21 2 1 1

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


29/42


33

Vy 10m tho yu cu bi ton .

Cch 2 :

( )'' 6 6 f x x = +

Nghim ca phng trnh ( )'' 0f x = l 1 1x = < . Do , hm s cho nghch bin trn khong( )1;1 khi v ch khi ( ) 2' 1 3.1 6.1 1 0 10 f m m = + + + .2.Hm s cho xc nh trn .

Ta c : ( )2' 4 1 1y mx m x m = + +

Hm s ng bin trn khong ( )2;+ khi v ch khi

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

( ) ( ) ( )2 2 4 14 1 4 1, 2; , 2;4 1xx x m x x m x x x+

+ + + + ++ +

Xt hm s ( ) ( ) ( )( )

( )( ) ( )2 2

2

2 2 14 1, 2; ' 0, 2;

4 1 4 1

x xxg x x g x x g x

x x x x

++= + = < +

+ + + +

nghch bin trn khong ( )2;+ v ( ) ( )2

9lim , lim 0

13 xxg x g x

+ += =

x 2 +( )'g x

( )g x 913 0

Vy9

13m tho yu cu bi ton .

V d 5:

Gii:



1. ( ) ( ) ( )3 2 22 7 7 2 1 2 3y x mx m m x m m = + + ng bin trn khong ( )2; + ?.2. ( )2 1 1

2

mx m x y

x m

+ + =

ng bin trn khong ( )1; + ?.


30/42


34

Ta c : ( ) ( )2 2' 3 2 2 7 7y x mx m m g x = + =

Hm s cho ng bin trn khong ( )2; + khi v ch khi ( )' 0, 2;y x +

Xt hm s ( ) ( )2 23 2 2 7 7g x x mx m m = + trn khong ( ) ( )2; ' 6 2x v g x x m + = Cch 1:Hm s ( )g x ng bin trn khong ( )2;+ khi v ch khi

( ) ( )2 2 25

2 0 3.2 2 .2 2 7 7 0 2 3 5 0 12

g m m m m m m + + + .

Vi cch gii ny hc sinh nn dng cho bi trc nghim, gc bi ton t lun thiu i tnh chun xcv trong sng ca bi ton .

Cch 2 :

( )' 0 3m

g x x= =

Nu 2 63

mm , khi ( ) ( )0, 2;g x x +

( )( ) 2

2;

5min 0 2 3 5 0 1

2xg x m m m

+ + +

Nu 2 63

mm> > , kh nng ny khng th xy ra (v sao ?).

2.Hm s cho xc nh trn \ 2

m

D

= .

Nu 0m = , ta c2

1 1' 0, 0

2 2

xy y x

x x

= = > . Hm s ng bin trn cc khong

( ) ( ); 0 0;v + , do cng ng bin trn khong ( )1;+ Vy ( )0m a= tho mn yu cu bi ton .Nu 0m , ta c

( )

( )

( )

( )2 2 2

2 2 2

2 2

2 2 2' , 2 2 2

2 2

g xmx m x m m y g x mx m x m m

x m x m

+= = = +

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

( )( )

( )2

2 00

1; 2 0 12

21 3 2 0 13

mm

mm m b

g m m m

> >

+ <

= + +


31/42


35

T ( ) ( )a v b suy ra 0 1m th tho mn yu cu bi ton .V d 6:

Gii :

Hm s cho xc nh trn .

Ta c : 2' 3 6y x x m = + + c'

' 9 3y

m =

Nu 3 ' 0 ' 0,g

m y x , khi hm s lun ng bin trn , do 3m

khng tho yu cu bi ton .

Nu 3 ' 0g

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

bin trong on1 2;x x vi di 2 1l x x=

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

2,3

mx x x x + = =

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

( ) ( )2 2

2 1 1 2 1 2

4 91 4 1 4 1

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

Cu hi nh : Tm tt c cc tham s m hm s 3 23y x x mx m = + + + nghch bin trn on c

di bng 1 .C hay khng yu cu bi ton tho :2 1

1?.l x x=

NG DNG N IU TRONG BT NG THC

V d 1:

Gii :

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

+ + + +

0x y z

Tm tt c cc tham s m hm s 3 23y x x mx m = + + + nghch bin trn on c di bng

1?.

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

+ + + +

2. Cho , , 0x y z > Chng minh rng:4 4 4 2 2 2 2 2 2

( ) ( ) ( ) ( )x y z xyz x y z xy x y yz y z zx z x + + + + + + + + + + .3. Chng minh rng : ln 1 , 0x x x> + >


32/42


36

Xt hm s : ( )x z y x y z

f xz y x y z x

= + + + +

.

Ta c: ( )2 2 2

1 1 1 1'( ) ( ) ( ) ( )( ) 0, 0

y z f x y z x f x

z y yz x x x= = l hm s ng bin

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

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

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

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

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

"( ) 0f x > (do x y z ) 2 3 2'( ) '( ) ( ) 0f x f y z y z z y z = = nn ( )f x l hm s ng

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

3. Chng minh rng : ( )ln 1 , 0x x x> + > Hm s ( ) ( )ln 1 f x x x = + xc nh v lin tc trn na khong )0; + v c o hm

( )1

' 1 01

f xx

= >+

vi mi 0x > . Do hm s ( )f x ng bin trn na khong )0; + , hn

na ( ) ( )0 0 f x f > = vi mi 0x >

Hay ( )ln 1 , 0x x x> + >

V d 2:

Gii :

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

a b c

a b b c c a + +

+ + +

t , , 1b c a

x y z xyz a b c

= = = = v bt ng thc cho1 1 1 3

1 1 1 2x y z + +

+ + +.

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

1 1 1 1

z

x y xy z+ =

+ + + +

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

a b c

a b b c c a + +

+ + +.

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

+ + +

+ + + +.


33/42


37

2

1 1 1 2 1 2 1( )

1 1 1 1 11 1

z tf t

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

+ + + + ++ +vi 1t z=

Ta c:2 2 2 2 2

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

2(1 ) (1 ) (1 )

t t f t f t f t

t t t

= =

+ + +pcm.

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

a b c c a

b c c a a b a c a + + +

+ + + +

t , ,1b c

x xa a

= = . Khi bt ng thc cn chng minh tr thnh

222 2 2 4 1 2 ( 1)1 (2 2 )

1 1 1 1

x x x x x x x x

x x x x

+ + + ++ + + + + +

+ + + + + +

Xt hm s 21 2 ( 1)

( ) 1 (2 2 ), 11

x x x f x x x x

x

+ += + + + +

+ +

Ta c: 2 22(2 1) 1 2 +1 2

'( ) 2 1 2 ( 1)[ ] 0, 11 +1( ) ( )

x x

f x x x x x

+ = + =

+ + +

Nh vy hm ( )f x l ng bin do 21

( ) ( ) 3 3 f x f

= +

Nhng 32 2 2

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

= + = + + =

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

BI TP T LUYN

1. Chng minh rng hm s ( ) 21 f x x = nghch bin trn on 0;1 .

2. Chng minh rng hm s ( ) 3 24

2 33

f x x x x = + ng bin trn .

3. Xt chiu bin thin ca cc hm s:

( ) 5 4 310 7

) 2 53 3

a f x x x x = + +

( ) 3 2) 2 1b f x x x x = + +

( )4

)c f x x

x

= +

( )9

)d f x x x

=

( ) 3 21

) 2 4 53

e f x x x x = +

( )2 8 9

)5

x x f f x

x

+=

( ) 2) 2 3g f x x x = +


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38

( )1

) 21

h f x x x

= +

( )) 3 1i f x x = +

( ) 2) 4 j f x x x =

( ))k f x x x = +

( ))l f x x x =

( ) 22

)9

xm f x

x=

4. Xt chiu bin thin ca cc hm s sau :

2

2

1 1)

21

)

33)

1) 2 3

a yx xx

b y

xx

c yx

d y x x

=

+=

=+

= + +

4 3

4 3 2

5 3

7 6 5

1) 5

23 3

) 2 6 114 2

4) 8

5 7) 9 7 12

5

e y x x x

f y x x x x

g y x x

h y x x x

= + +

= + +

= + +

= + +

5. Chng minh rng :

)a Hm s2

2

xy

x

=

+ng bin trn mi khong xc nh ca n .

)b Hm s2 2 3

1

x xy

x

+=

+nghch bin trn mi khong xc nh ca n .

6. Chng minh rng :

)a Hm s

=

+

3

1 2

xy

xnghch bin trn mi khong xc nh ca n .

)b Hm s+

=+

22 3

2 1

x xy

xng bin trn mi khong xc nh ca n .

)c Hm s = + +2 8y x x nghch bin trn .

)d Hm s = + 2cosy x x ng bin trn .7. Chng minh rng :

)a Hm s = 22y x x nghch bin trn on 1;2


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40

)b Chng minh rng vi mi ( ) 1;1m , phng trnh + =2sin cosx x mc nghim duy nht thuc

on 0; .

0;

3x ta c ( )

50 1

3 4y y y y nn phng trnh cho khng c nghim ( ) 1;1m

;3

x ta c ( )

51

3 4y y y y . Theo nh l v gi tr trung gian ca hm s

lin tc vi ( )

51;1 1;

4m , tn ti mt s thc

;

3c sao cho ( ) = 0y c . S cl nghim

ca phng trnh + =2sin cosx x m v v hm s nghch bin trn on

;3

nn trn on ny ,

phng trnh c nghim duy nht .

Vy phng trnh cho c nghim duy nht thuc on 0; .

10. Cho ( ) ( )1;1 , 2;4A B l hai im ca parabol = 2y x .Xc nh im C thuc parabol sao cho tiptuyn ti C vi parabol song song vi ng thng AB .

11. Vi gi tr no ca ahm s ( ) 3 f x x ax = + nghch bin trn .12. Vi gi tr no ca m, cc hm s ng bin trn mi khong xc nh ca n ?

) 21

ma y x

x= + +

( )22 2 3 1)

1

x m x m b y

x

+ + +=

Hng dn :

( )= + + =

2

) 2 ' 1 , 11 1

m ma y x y x

x x

0m th > ' 0; 1y x . Do hm s ng bin trn mi khong ( );1 v ( )+1; .

> 0m th( )

( )

( )

= =

2

2 2

1' 1 , 1

1 1

x mmy x

x x

v = = ' 0 1y x m . Lp bng bin thin ta

thy hm s nghch bin trn mi khong

( )1 ;1m v

( )+1;1 m ; do khng tho iu kin .

Vy :hm s ng bin trn mi khong xc nh ca n khi v ch khi 0m

Ch : Bi ton trn c m rng nh sau

1)a Tm gi tr ca m hm s ng bin ( ) ; 1

2)a Tm gi tr ca m hm s ng bin ( )+2;

3)a Tm gi tr ca m hm s nghch bin trong khong c di bng 2.


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41

4)a Tm gi tr ca m hm s nghch bin trn mi khong ( )0;1 v ( )1;2 .

5)a Gi phng trnh ' 0y = c hai nghim 1 21x x< < hm s ng bin trn mi khong

( ) ( )1 2;1 1;x v x , trng hp ny khng tha .13. Vi gi tr no ca m, cc hm s nghch bin trn

( )3 21

2 2 1 3 23

y x x m x m = + + + +

Hng dn :

( )3 2 21

2 2 1 3 2 ' 4 2 1, ' 2 53

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

=

5

2m th ( )= 2

' 2 0y x vi mi =, ' 0x y ch ti im = 2x . Do hm s nghch bintrn .

( ) < >5 ' 02

m hay th =' 0y c hai nghim ( )


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42

3)a Tm gi tr ca m hm s ng bin trong khong c di bng 1.

4)a Tm gi tr ca m hm s nghch bin trn khong ( )0;1 .

14. Cho hm s ( ) ( ) ( )3 21 2

1 2 33 3

f x x m x m x = + +

)a

Vi gi tr no ca m, hm s ng bin trn

)b Vi gi tr no ca m, hm s ng bin trn :( )1) 1;b +

( )2) 1;1b

(3) ; 1b

4) 1;0b

15. Cho hm s ( ) 2 sin tan 3 f x x x x = +

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

.

)b Chng minh rng 2 sin tan 3x x x+ > vi mi 0;2

x

.

Hng dn :

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

Hm s ( ) 2 sin tan 3 f x x x x = + lin tc trn na khong 0;2

v c o hm

( )( ) ( )

23 2

2 2 2

1 cos 2 cos 11 2 cos 1 3 cos' 2 cos 3 0, 0;

2cos cos cos

x xx x f x x x

x x x

+ + = + = = >

Do hm s ( ) 2 sin tan 3 f x x x x = + ng bin trn na khong 0;2

)b Chng minh rng 2 sin tan 3x x x+ > vi mi 0;2

x

Hm s ( ) 2 sin tan 3 f x x x x = + ng bin trn na khong 0;2

v

( ) ( )0 0, 0;2

f x f x

=

; do 2 sin tan 3 0x x x+ > mi 0;2

x

hay

2 sin tan 3x x x+ > vi mi 0;2

x

16.


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43

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

x

.

)b Chng minh rng3

tan3

xx x> + vi mi 0;

2x

.

Hng dn :)a Chng minh rng hm s ( ) tan f x x x = ng bin trn na khong 0;

2

.

Hm s ( ) tan f x x x = lin tc trn na khong 0;2

v c o hm

( ) 221

' 1 tan 0, 0;2cos

f x x x x

= = >

.

Do hm s ( ) tan f x x x = ng bin trn na khong 0;2

v ( ) ( )0 0, 0;

2 f x f x

> =

hay tan x x> .

)b Chng minh rng3

tan3

xx x> + vi mi 0;

2x

.

Xt hm s ( )3

tan3

xg x x x = trn na khong 0;

2

.

Hm s ( )3

tan3

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

2

v c o hm

( ) ( ) ( )2 2 2

2

1' 1 tan tan tan 0, 0;

2cosg x x x x x x x x x

x

= = = + >

cu )a

Do hm s ( )3

tan3

xg x x x = ng bin trn na khong 0;

2

v

( ) ( )0 0, 0;2

g x g x

> =

hay3

tan3

xx x> + vi mi 0;

2x

.

17. Cho hm s ( )4

tan f x x x

= vi mi 0;4

x

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

.

)b T suy ra rng4

tanx x

vi mi 0;4

x

.

Hng dn :

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

.


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45

Vy2

cos 12

xx > vi mi 0x

)c Hm s ( )3

sin6

x f x x x = . Theo cu b th ( )' 0, 0 f x x < . Do hm s nghch bin trn .

V ( ) ( )( ) ( )0 0

0 0

f x f khi x

f x f khi x

>

)d sin tan 2x x x+ > vi mi 0;2

x

Hm s ( ) sin tan 2 f x x x x = + lin tc trn na khong 0;2

v c o hm

( ) 22 21 1

' cos 2 cos 2 0, 0;2cos cos

f x x x x x x

= + > + >

. Do hm s ng bin trn na

khong 0; 2

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

> =

MT S DNG TON TRONG K THI T TI &TUYN SINH IHC

1 Tm tham s m th ca hm s ng bin trn :

)a 3 21 (3 2)3

my x mx m x

= + +

)b ( )3 21 2 1 13y x x m x = + + 2 Tm m cc hm s sau nghch bin trn

( ) ( )3 21

2 2 2 2 53

my x m x m x

= + +

3 Tm m cc hm s sau ng bin trn )a siny x m x = + )b 1 1sin sin2 sin 3

4 9y mx x x x = + + +

)c 2 212 2cos sin cos cos 24

y mx x m x x x = +

)d ( 3) (2 1)cosy m x m x = + 4 Tm tham s m th ca hm s :

)a 3 23 ( 1) 4y x x m x m = + + + + nghch bin trn ( )1;1 )b 3 2 2(2 7 7) 2( 1)(2 3)y x mx m m x m m = + + ng bin trn )2; +


42/42


)c 3 23y x x mx m = + + + nghch bin trn on c di bng 1.)d 2(2 1) 3 5

1

m x mx y

x

+=

ng bin trn on 2;5

)e 22 32 1

x x my

x

+=

+nghch bin trn khong

1;

2

+

)f 2 88( )

x xy

x m

=

+ng bin trn khong ( )1; +

)g 21

mx x m y

mx

+ +=

+ng bin trn khong ( )0;+ .

5 Chng minh rng :

)a sin tan 12 2 2 , 0;2

x x x x+ + >

)b 22

1 cos , 0;

4 4

x x x +

< <

)c 0 05 tan6 6 tan5>

)d 2009 20082008 2009>

)e2 2

tan tan , 02cos cos

a b a ba b a b

b a

< < < < <

6 Chng minh rng :

)a ln , 0b a b b a

a ba a b

> > < <

)b ( )

1

lg lg 41 10 1;0 1,

y x

y x y x

x y x y

>

< < < <

)c , 0, 0,ln ln 2

a b a bab a b a b

a b

+< < > >

)d1

lg ( 1) lg ( 2), 1x x

x x x+

+ > + >

)e , 02 ln ln

x y x y x y

x y

+ > > >

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ÔN THI ĐẠI HỌC - TUYỆT CHIÊU HÀM SỐ 12 -2009