2.1. tinh don dieu cua ham so

Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12

Tnh n iu ca hm s

5

TM TT L THUYT 1. nh ngha : Gi s K l mt khong , mt on hoc mt na khong . Hm s f xc nh trn K c gi l

ng bin trn K nu vi mi ( ) ( )1 2 1 2 1 2, ,x x K x x f x f x < < Nghch bin trn 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 s hm 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 ti mi 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 bin trn ;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 nghch bin trn ;a b

TNH N IU CA HM S


Tnh n iu ca hm s

6

CC BI TON C BN V d 1: Xt chiu bin thin ca cc hm s : Gii :

( ) 3 21) 3 8 23

a f x x x x= +

Hm s cho xc nh trn .

Ta c ( ) 2' 6 8f x x x= + ( )' 0 2, 4f 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 xf 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 23

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= +


Tnh n iu ca hm s

7

Vy hm s ng bin trn mi khong ( );1 v ( )1;+ ( ) 3 2) 3 3 2c f x x x x= + + +


Ta c ( ) ( )22' 3 6 3 3 1f x x x x= = + = + ( )' 0 1f 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 6f x x x= + ( ) ( ) ( ) ( )' 0, ; 1 , 0;f x x f x> + ng bin trn mi khong ( ); 1 v ( )0;+ . ( ) ( ) ( )' 0, 1;0f 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 kt lun.

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=


Tnh n iu ca hm s

8

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

Ta c ( ) 3' 4 4f x x x= ( ) ( ) ( ) ( )' 0, 1;0 , 1;f x x f x> + ng bin trn mi khong ( )1;0 v ( )1;+ . ( ) ( ) ( ) ( )' 0, ; 1 , 0;1f 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 ri kt lun.

( ) 3 24 2) 6 93 3

c f x x x x= +


Ta c ( ) ( )22' 4 12 9 2 3f x x x x= + =

( ) 3' 02

f x x= = v ( )' 0f x < vi mi 32

x

V hm s nghch bin trn mi na khong 3;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

xf x x

x x

=

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

Hoc c th trnh by :

( ) ( ) ( )' 0, 0;1f x x f x> ng bin trn on 0;1 ( ) ( ) ( )' 0, 1;2f 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;

2f 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 :

t 2sin ; 0 1t x t= .

Khi phng trnh ( ) 5 5 81* 81 (1 ) , 0;1256

t t t + =

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

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

4 481 (1 ) 1

'( ) 040;1

t tf 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 nghim 21 1 1sin cos2 ( )

4 4 2 6t x x x k k Z

= = = = + .

Chng minh rng : sin tan 2 , 0;2

x x x x

+ >

.

Gii phng trnh : ( )10 10 81 81sin cos *256

x x+ =


Tnh n iu ca hm s

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 , 0f t t t t t= + + >

Ta c ( )3

4 2

2 3'( ) 2 0, 0

3

t tf 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 = = = + =

Vy 1

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( ) osxf x e c x= + lin tc trn khong - ;

2 2x

. 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= + + +


Tnh n iu ca hm s

12

23

2

3

tantan

2

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

cos os

xx c xf x x e x x

x c x

= =

V 2

3tan2 2 os 0xe c x > > Nn du ca '( )f x chnh l du ca sinx . T y ta c ( ) (0) 2f 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 xf x x= +

Ta c: '( ) 2003 ln2003 2005 ln2005 4006x xf x = + 2 2''( ) 2003 ln 2003 2005 ln 2005 0 "( ) 0 x xf 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 , 0f t t t t= + > ta c ( ) ( )1' 1 0, 0ln 3

f t t f tt

= + > > l hm ng bin

khong ( )0;+ nn phng trnh ( ) ( )* (3 ) (1 2 ) 3 2 1 3 2 1 0 * *x x xf f x x x = + = + = Xt hm s: 2( ) 3 2 1 '( ) 3 ln 3 2 "( ) 3 ln 3 0x x xf x x f x f x= = = >


Tnh n iu ca hm s

13

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

uu u u

+ + = + + =

Xt hm s : ( ) ( ) 231

log 2 .55

uf u u

= + +

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

( )2' 1 1( ) 5 .ln 5.2 0, 0( 2)ln 3 5

uf u u u f uu

= + > +

ng bin trn na khong )0; + v

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

Khi 2 23 5

23 2 1 3 1 03 5

2

xx 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

3

2 1

2 2

x x y

y y x

+ = + =

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =


Tnh n iu ca hm s

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 , ; 42

f t t t t = +

, ta c:

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

f x tt 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 yx 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

= + + = + = =


Tnh n iu ca hm s

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 02 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 nht 0

0

x

y

= =

.

3.

3 3

6 6

3 3 (1)

1 (2)

x x y y

x y

=

+ =

T (1) v (2) suy ra 1 , 1x y (1) ( ) ( ) (*)f x f y =

Xt hm s 3( ) 3f 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:


Tnh n iu ca hm s

16

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 bit 1 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

= = +


Tnh n iu ca hm s

17

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 + + =

Xt hm s 4 / 33

1( ) 2 ( ) 4 1 0 .

4f x x x f x x x

= + + = + = =

4

3 3

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

4 4 4 x xf 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

+ + + = +

+ + + + =


Tnh n iu ca hm s

18

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 li 2

2

1 3( )

1 3

v

u

u uII

v v

+ + = + + =

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

( ) ( )2

2 2 2

11 0,

1 1 1

x xx x xf 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)

II u 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 45[( ) ( ) ] 1 2.2 *5 5

t t t+ = +

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

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

D thy : ( ) 1 45[( ) ( ) ]5 5

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

v ( ) ( )1 1 5 1f 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 + + + + + =


Tnh n iu ca hm s

19

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

22 2

2 2

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

1 1

y y yf 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 tt

= =

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

( ) ( )' 0, 1tf 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 yy f x g y f y g xf y g xx

ex

= + = + = +

+ = =

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

Khi ( ) ( )2 22

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

20071 1

20071

x

y

ye

y

xe

x

=

=

c ng 2 nghim tha mn iu kin

1, 1x y> >


Tnh n iu ca hm s

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

12

1

xy

xy

zy

zx

z

=

=

=

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 )

tf 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 0

9 27 27 0

y x x

z y y

x z z

+ =

+ = + =


Tnh n iu ca hm s

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 27f t t t f t t= + =

( )

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

32' 0,

2

f t tf 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 s 3 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

+ + + =

+ + + =


Tnh n iu ca hm s

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 s 3 2( ) 3 3 ln( 1),f t t t t t t R= + + + .

Ta c: ( )22

2 1'( ) 3 3 0,

2 1

tf 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

+ =

+ =

+ =


Tnh n iu ca hm s

23

H cho

32

32

32

log (6 )2 6 ( ) ( )

log (6 ) ( ) ( )2 6 ( ) ( )

log (6 )2 6

xy

x x f y g xy

z f z g y

y y f x g zz

x

z z

= + =

= = + =

= +

Xt hm s 3

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

2 6

tf t t g t t

t t

= = +

Ta c ( ) ( )

1'( ) 0, ( ;6)

6 ln 3f t t f t

t= <

nghch bin trn khong ( ;6) v

( )( )

32

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 g l cc hm cng tng hoc cng gim trn A v 1 2

( , ,..., )n

x x x l nghim ca h

trn A th 1 2

...n

x x x= = =

nh l 2:Nu ,f g khc tnh n iu trn A v 1 2

( , ,..., )n

x 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) 5

x y x y

x y

=

+ =


Tnh n iu ca hm s

24

Gii :

1.

sin sin 3 3 (1)

(2) 5

, 0 (3)

x y x y

x y

x y

=

+ =

>

T ( ) ( )2 , 3 , (0; )5

x y

( ) ( )1 sin 3 sin 3 *x x y y = .

Xt hm s ( ) sin 3 , (0; )5

f t t t t

= ta c ( ) ( )' cos 3 0, (0; )5

f t t t f t

= < l hm

nghch bin trn khong (0; )5

t

nn ( ) ( ) ( )* f x f y x y = =

Vi x y= thay vo ( )2 ta tm c 10

x y

= =

Vy ( ); ;10 10

x y

=

l nghim ca h.

2.2 3

2 3

log (1 3cos ) log (sin ) 2

log (1 3 sin ) log (cos ) 2

x y

y x

+ = +

+ = +

iu kin : cos 0

sin 0

x

y

>

>


Tnh n iu ca hm s

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 ) logf 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 1f x x x= + lin tc trn na khong )0; + , ta c

( )( )324

1 1' 0

21

xf x

xx

=


Tnh n iu ca hm s

26

iu kin: 3 1x .

Phng trnh 3 3 4 1 1

(1)4 3 3 1 1

x xm

x x

+ + + =

+ + +

Nhn thy rng: ( ) ( )2 2

2 2 3 13 1 4 1

2 2

x xx x

+ + + = + =

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

=

sao cho:

2

23 2 sin 2

1

tx

t+ = =

+ v

2

2

11 2 cos 2

1

tx

t

= =

+

( ) ( )2

2

3 3 4 1 1 7 12 9, 2

5 16 74 3 3 1 1

x x t tm m f t

t tx x

+ + + + += = =

+ ++ + +

Xt hm s: 2

2

7 12 9( )

5 16 7

t tf t

t t

+ += + +

lin tc trn on 0;1t . Ta c

( )( )

2

22

52 8 60'( ) 0, 0;1

5 16 7

t tf t t f t

t t

= < + +

nghch bin trn on [ ]0;1 v 9 7(0) ; (1)7 9

f 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 +


Tnh n iu ca hm s

27

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

5

+

, ta c

( )5 1 1'( ) 0 ,52 5 1 2 1

f x x f xx 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 . 5

2. 3 3 2 2 62 1

x xx

+

V d 2:

Gii:

iu kin: 1 3

2 2x<

Bt phng trnh cho 5

3 3 2 2 6 ( ) ( ) (*)2 1

x x f x g xx

+ +

Xt hm s 5

( ) 3 3 22 1

f x xx

= +

lin tc trn na khong 1 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 on 1 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 +


Tnh n iu ca hm s

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 xf x x x f x

x x x

+ + += + + > >

+ + ng

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

Vy nghim ca bt phng trnh cho l: 1

72x .

V d 3:

Gii :

iu kin:

3 22 3 6 16 02 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 s

3 2( ) 2 3 6 16 4f 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 xf x x f x

xx x x

+ += + >

+ + + ng bin trn na

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


Tnh n iu ca hm s

29

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

CC BI TON LIN QUAN THAM S V d 1: Gii: 1. Hm s cho xc nh trn .

Ta c ( ) 2' 2 4f 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 2f x x x ax x a a hay a + +

Cch 2 :

2 4a = Nu 2 4 0 2 2a hay a < < < th ( )' 0f x > vi mi x . Hm s ( )f x ng bin trn Nu 2a = th ( ) ( )2' 2f x x= + ( ) ( )' 0 2, ' 0, 2f x x f x x= = > . Hm s ( )f x ng bin trn 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 s nghch bin trn khong ( )1 2;x x ,ng bin trn mi khong ( )1;x v ( )2;x + . Do 2a < hoc

2a > khng tho mn 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 3f x a x a x g x= + + + =

Vi gi tr no ca a hm s sau ng bin trn .

( ) 3 211. 4 33

f x x ax x= + + +

( ) ( ) ( )2 3 212. 1 1 3 53

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


Tnh n iu ca hm s

30

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

( ) ( ) 31 ' 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 xf 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 m hm s ( ) ( )21 2 1

1

m x xf x

x

+ +=

+ ng bin mi khong xc nh .

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

+=

+ nghch bin khong ( );1 .


Tnh n iu ca hm s

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 2

2 11 1;1

m m mm

m mm

< < < < .

Xt hm s ( ) 26 4g x x x= lin tc trn khong ( )1;+ , ta c ( ) ( )' 12 4 0, 1g x x x g x= > > ng bin trn 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 m sao 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 ?.


Tnh n iu ca hm s

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 0

4lim , lim

9x xg x g x

+ = =

x 3 0

( )'g x

( )g x 49

Da vo bng bin thin suy ra 4

9m

V d 4 : Gii : 1. Hm s cho xc nh trn .

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

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

( )( ) ( )2

1;13 6 1 , 1;1 min 1

xm 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 1lim 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 13

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


Tnh n iu ca hm s

33

Vy 10m tho yu cu bi ton . Cch 2 :

( )'' 6 6f 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 10f 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 24 1

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

xx x m x x m x

x x

+ + + + + +

+ +

Xt hm s ( ) ( ) ( ) ( )( )

( ) ( )2 22

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

Vy 9

13m tho yu cu bi ton .

V d 5: Gii: 1. Hm s cho xc nh trn .


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 12

mx m xy

x m

+ + =

ng bin trn khong ( )1;+ ?.


Tnh n iu ca hm s

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 2 52 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 xc v trong sng ca bi ton . Cch 2 :

( )' 03

mg 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

mD

=

.

Nu 0m = , ta c 2

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 22 2 2

2 2

2 2 2' , 2 2 2

2 2

g xmx m x m my 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

> >

+ <

= + +


Tnh n iu ca hm s

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 on

1 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 22 1 1 2 1 24 9

1 4 1 4 13 4

x 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 :

1 Cho 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> + >


Tnh n iu ca hm s

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 zf x y z x f x

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

0x ( ) ( ) 0f 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 ( ) 2f 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 1f 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 0f 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 xyza b c

= = = = v bt ng thc cho 1 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 ( )

3( )

a b c c a

b c c a a b a c a

+ + +

+ + + +.


Tnh n iu ca hm s

37

2

1 1 1 2 1 2 1( )

1 1 1 1 11 1

z tf t

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

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

Ta c: 2 2 2 2 2

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

2(1 ) (1 ) (1 )

t tf 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 xx x

x x x x

+ + + +

+ + + + + ++ + + + + +

Xt hm s 21 2 ( 1)

( ) 1 (2 2 ), 11

x x xf x x x x

x

+ +

= + + + + + +

Ta c: 2 2

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

1 +1( ) ( )

x xf x x x

x x

+ = + =

+ + +

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

( ) ( ) 3 3f x f

= +

Nhng 32 2 2

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

= + = + + =

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

BI TP T LUYN

1. Chng minh rng hm s ( ) 21f 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 xx

= +

( ) 9)d f x xx

=

( ) 3 21) 2 4 53

e f x x x x= +

( )2 8 9

)5

x xf f x

x

+=

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

1Highlight

1Highlight


Tnh n iu ca hm s

38

( ) 1) 21

h f x xx

= +

( )) 3 1i f x x= + ( ) 2) 4j 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 yxx

c yx

d y x x

=

+=

=+

= + +

4 3

4 3 2

5 3

7 6 5

1) 5

23 3

) 2 6 114 24

) 85

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 s 2

2

xyx

=

+ ng bin trn mi khong xc nh ca n .

)b Hm s 2 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

x nghch bin trn mi khong xc nh ca n .

)b Hm s +

=+

22 3

2 1

x xy

x ng 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


Tnh n iu ca hm s

39

)b Hm s = 2 9y x ng bin trn na khong ) +3;

)c Hm s = +4

y xx nghch bin trn mi na khong ) 2;0 v ( 0;2

)d Hm s 2 1

xyx

=+

ng bin trn khong ( )1;1 , nghch bin trn mi khong ( ); 1 v

( )1;+ . 8. Cho hm s = 22 2y x x

)a Chng minh hm s ng bin trn na khong ) +2; )b Chng minh rng phng trnh =22 2 11x x c nghim duy nht .

Hng dn :

)a( ) ( )

= > +

5 8' 0, 2;

2

x xy x

x. Do hm s ng bin trn na khong ) +2;

)b Hm s xc nh v lin tc trn na khong ) +2; , do cng lin tc trn on 2;3 , ( ) ( )2 11 3y y< < nn theo nh l gi tr trung gian ca hm s lin tc, tn ti s thc ( ) 2;3c sao

cho ( ) = 11y c . S thc ( ) 2;3c l 1 nghim ca phng trnh cho v v hm s ng bin trn na khong ) +2; nn ( ) 2;3c l nghim duy nht ca phng trnh . 9. Cho hm s = +2sin cosy x x .

)a Chng minh rng hm s ng bin trn on

0;3

v nghch bit trn on

;3

.

)b Chng minh rng vi mi ( ) 1;1m , phng trnh + =2sin cosx x m c nghim duy nht thuc on 0; .

Hng dn :

)a Chng minh rng hm s ng bin trn on

0;3

v nghch bit trn on

;3

.

Hm s lin tc trn on 0; v ( ) ( )= ' sin 2 cos 1 , 0;y x x x

V ( )0; sin 0x x > nn trong khong ( ) ( ) 10; : ' 0 cos2 3

f x x x

= = =

>

' 0, 0;

3y x nn hm s ng bin trn on

0;3

<

' 0, ;

3y x nn hm s nghch bin trn on

;3


Tnh n iu ca hm s

40

)b Chng minh rng vi mi ( ) 1;1m , phng trnh + =2sin cosx x m c 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

;

3x 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

;3

c sao cho ( ) = 0y c . S c l 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 tip tuyn ti C vi parabol song song vi ng thng AB .

11. Vi gi tr no ca a hm s ( ) 3f 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 mb 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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Tnh n iu ca hm s

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 bin

trn .

( ) < < 5 ' 02

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

( ) > > 5 ' 02

m hay th =' 0y c hai nghim ( )< 1 2 1 2,x x x x . Hm s ng bin trn khong

( ) 1 2;x x . Trng hp ny khng tha mn . Ngoi ra ta c th trnh by : Hm s nghch bin trn khi v ch khi

1 0 52 5 0

' 0 2

am m

=


Tnh n iu ca hm s

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 21 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 3f 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 3f x x x x= + lin tc trn na khong 0;2

v c o hm

( ) ( ) ( )2

3 2

2 2 2

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

2cos cos cos

x xx xf x x x

x x x

+ + = + = = >

Do hm s ( ) 2 sin tan 3f 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 3f 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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Tnh n iu ca hm s

43

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

x

.

)b Chng minh rng 3

tan3

xx x> + vi mi 0;

2x

.

Hng dn :

)a Chng minh rng hm s ( ) tanf x x x= ng bin trn na khong 0;2

.

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

v c o hm

( ) 221

' 1 tan 0, 0;2cos

f x x xx

= = >

.

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

v ( ) ( )0 0, 0;

2f x f x

> =

hay tanx x> .

)b Chng minh rng 3

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 2 221

' 1 tan tan tan 0, 0;2cos

g x x x x x x x x xx

= = = + >

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

> =

hay 3

tan3

xx x> + vi mi 0;

2x

.

17. Cho hm s ( ) 4 tanf x x x

= vi mi 0;4

x

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

.

)b T suy ra rng 4

tanx x

vi mi 0;4

x

.

Hng dn :

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

.


Tnh n iu ca hm s

44

Hm s ( ) 4 tanf x x x

= lin trc trn on 0;4

v c o hm

( ) ( ) 224 1 4 4

' tan , 0; , ' 0 tan4cos

f x x x f x xx

= = = =

V 4

0 1 tan4

< < = nn tn ti mt s duy nht 0;4

c

sao cho 4

tanc

=

( ) ( ) ' 0, 0;f x x c > hm s ( )f x ng bin trn on 0;x c

( ) ' 0, ;4

f x x c

<

hm s ( )f x nghch bin trn on ;4

x c

)b D thy ( ) ( ) 4 40 ; 0; tan 0 tan4

f x f c x x x hay x x

vi mi 0;4

x

.

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

)b 2

cos 12

xx > vi mi 0x

)c3

sin6

xx x> vi mi 0x > ,

3

sin6

xx x< vi mi 0x <

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

x

Hng dn : )a sinx x< vi mi 0x > .

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

v c o hm

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

xf x x x

= = >

. Do hm s ng bin trn na khong 0;

2

v ta c

( ) ( )0 0, 0;2

f x f x

> =

, tc l sin 0, 0; sin , 0;2 2

x x x hay x x x

> >

.

)b 2

cos 12

xx > vi mi 0x

Hm s ( )2

cos 12

xf x x= + lin tc trn na khong )0; + v c o hm ( )' sin 0f x x x= >

vi mi 0x > ( theo cu a ). Do hm s ( )f x ng bin trn na khong )0; + v ta c

( ) ( )0 0, 0f x f x> = > , tc l 2

cos 1 0, 02

xx x + > >

Vi mi 0x < , ta c ( ) ( )2

2

cos 1 0, 0 cos 1 0, 02 2

x xx x hay x x

+ > < + > <


Tnh n iu ca hm s

45

Vy 2

cos 12

xx > vi mi 0x

)c Hm s ( )3

sin6

xf x x x= . Theo cu b th ( )' 0, 0f 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 2f 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 xx x

= + > + >

. Do hm s ng bin trn na

khong 0;2

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

2f x f x

> =

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

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 13

y 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 1

sin sin2 sin 34 9

y mx x x x= + + +

)c 2 21

2 2 cos 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; +


Tnh n iu ca hm s

46

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

)d 2(2 1) 3 5

1

m x mxy

x

+=

ng bin trn on 2;5

)e 22 3

2 1

x x my

x

+=

+ nghch bin trn khong

1;

2

+

)f 2 8

8( )

x xy

x m

=

+ ng bin trn khong ( )1;+

)g 2

1

mx x my

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 tan 5>

)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

( )

1lg lg 41 1

0 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 xx x x

++ > + >

)e , 02 ln ln

x y x yx y

x y

+ > > >

2.1. tinh don dieu cua ham so

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