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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
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
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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 =
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
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( ) 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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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Vy h phng trnh c 2 nghim phn bit
113
9,3 11
9
xx
yy
= =
= =
.
2.
( )
( )
3
3
2 1
2 2
x x y
y y x
+ = + =
Cch 1 :
Xt hm s 3 / 2( ) 2 ( ) 3 2 0, f t t t f t t t = + = + > .
H phng trnh 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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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.
Vy h phng trnh c 3 nghim phn bit
1 5 1 51
2 21 1 5 1 5
2 2
x xx
yy y
+ = = = = + = =
.
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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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
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
+ =
+ = + =
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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
+ + + = + + + =
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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
+ =
+ =
+ =
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 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
=
+ =
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 ) 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
=
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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 +
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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 .
Gii cc bt phng trnh sau :
52. 3 3 2 2 6
2 1x x
x +
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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 < < .
Gii cc bt phng trnh sau :
3. ( 2)(2 1) 3 6 4 ( 6)(2 1) 3 2x x x x x x + + + + +
Gii cc bt phng trnh sau :
3 24. 2 3 6 16 2 3 4x x x x + + + < +
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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 = + + + +
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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 .
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 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 ?.
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 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 :
1.Hm s cho xc nh trn .
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
Tm iu kin ca tham s msao cho hm s :
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;+ ?.
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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.Hm s cho xc nh trn .
Tm iu kin ca tham s msao cho hm s :
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; + ?.
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 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
> >
+ <
= + +
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh 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 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> + >
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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
+ + +
+ + + +.
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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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( )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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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)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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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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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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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Nguyn Ph Khnh - Lt Cc vn lin quan Hm s lp 12Tnh n iu ca hm s
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)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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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; +
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)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
+ > > >