skkn_kythuatgiambientimgtnngtln

K thut gim bin trong bi ton tm GTNN GTLN ca mt biu thc

Gio vin: Trn Phi Thon 1

Bi ton tm gi tr nh nht (GTNN), gi tr ln nht (GTLN) ca mt biu

thc nhiu bin l mt bi ton bt ng thc v y l mt trong nhng dng ton kh

chng trnh ph thng. Trong thi tuyn sinh i hc, Cao ng hng nm, ni

dung ny thng xut hin dng cu kh nht. Trong Sch gio khoa Gii tch 12 th

ch trnh by cch tm GTNN, GTLN ca hm s (tc biu thc mt bin s). V vy,

mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha mt bin tr nn n

gin. Tuy nhin thc t, hu ht hc sinh l khng gii quyt c cho bi ton t hai

bin tr ln, thm ch cn c tm l khng c n.

Qua qu trnh ging dy lp chuyn Ton v luyn thi i hc ti tch ly

c mt s kinh nghim cho ni dung ny. Cc vn trnh by trong sng kin kinh

nghim l chuyn c ng dng trong ging dy lp 11T2 chuyn Ton ca

trng THPT chuyn Thoi Ngc Hu v cc lp luyn thi i hc. Sng kin kinh

nghim ny l s tng kt c chn ln cc chuyn ca bn thn vit ra trong thc

tin ging dy cng vi s ng gp nhit tnh ca qu Thy, C trong T Ton Tin

trng THPT chuyn Thoi Ngc Hu.

ti ny xut pht t nhng l do sau:

Gip hc sinh c thm kin thc v t tin hn trong vic gii quyt bi ton kh

ny.

Gip cho qu Thy, C v cc bn ng nghip dy Ton c mt ti liu tham

kho trong qu trnh ging dy b mn ca mnh. V qua chuyn ny ti hy

vng qu Thy, C v cc bn ng nghip s yu thch hn trong vic ging dy

chuyn ny. Thc t mt s Thy, C khng thch dy, v k c nhng Thy,

C nhiu nm luyn thi i hc cng khng i su lm v chuyn ny.

Phn m u

1. Bi cnh ca ti

2. L do chn ti



- ti ny c th p dng rng ri cho tt c gio vin dy Ton cc trng trung

hc ph thng tham kho v cc em hc sinh lp 12 n thi i hc, Cao ng.

- Phm vi nghin cu ca ti ny bao gm:

+ Nhc li cch tm GTNN, GTLN ca hm s thng qua mt vi v d.

+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin bng cch th mt bin qua bin cn li. + H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin bng cch t n ph theo tnh i xng t x y= + , 2 2t x y= + hoc t xy= . + H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai

bin bng cch t n ph theo tnh ng cp xty

= ..

+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha ba bin bng cch t n ph hoc th hai bin qua mt bin cn li.

Bn thn nghin cu ti ny nhm mc ch:

- Chia s vi qu Thy, C, cc bn ng nghip v cc em hc sinh kinh nghim

gii quyt bi ton tm GTNN, GTLN trong thi tuyn sinh i hc.

- Bn thn nhm rn luyn chuyn mn nhm nng cao nghip v s phm.

- Hng ng phong tro vit sng kin kinh nghim ca trng THPT chuyn Thoi

Ngc Hu.

Sng kin c chia thnh ba phn :

Phn m u Phn ni dung: gm 3 chng

Chng 1. Gi tr nh nht, gi tr ln nht ca hm s Chng 2. K thut gim bin trong bi ton tm GTNN, GTLN ca biu thc Chng 3. Mt s bi ton trong cc thi tuyn sinh i hc

Phn kt lun

3. Phm vi v i tng nghin cu

4. Mc ch nghin cu

5. Cu trc SKKN



Chng I

GI TR NH NHT , GI TR LN NHT CA HM S

Trong mc ny chng ti trnh by li mt s kin thc v o hm v mt s

cng thc v o hm.

1.1 nh l. Gi s D l mt khong hay hp cc khong.

Nu hai hm s ( )u u x= v ( )v v x= c o hm trn D th

( ) ;u v u v + = + ( ) ;u v u v =

( ) ;uv u v uv = + ( ) ;ku ku =

( ) 2u u v uvv v = , vi ( ) 0v x

1.2.nh l. o hm ca mt s hm s thng gp

( ) 0c = (c l hng s)

( ) 1x =

( ) ( )1n nx nx x = ( ) 1n nu nu u =

( ) 21 1x x = ( ) 21 uu u

=

( ) ( )12 0xx x = > ( ) 2u uu

=

( )x xe e = ( )u ue e u =

( ) ( )1ln 0xx x = > ( )ln uuu =

( )sin cosx x = ( )sin cosu u u =

( )cos sinx x = ( )cos sinu u u =

( ) ( )2 2tan 1 tanx x x k = + + ( ) ( )2tan 1 tanu u u = +

( ) ( )( )2t 1 cotco x x x k = + ( ) ( )2t 1 tco u u co u = +

Phn ni dung

I.1. Mt s kin thc c s v o hm



1.3 Nhn xt. o hm ca mt s hm phn thc hu t thng gp

1. Cho hm s ax bycx d

+=

+ vi . 0, 0a c ad cb . Ta c

( )2ad cb

cx dy

+ = .

2. Cho hm s 2ax bx c

ymx n+ +

=+

vi . 0a m . Ta c ( )

2

2

2b c

amx anxm n

mx ny

+ +

+ = .

3. Cho hm s 2

2

ax bx cy

mx nx p+ +

=+ +

vi . 0a m . Ta c ( )

2

22

2a b a c b c

x xm n m p n p

mx nx py

+ +

+ + = .

Trong mc ny chng ti trnh by li mt s kin thc v bi ton tm gi tr nh

nht, gi tr ln nht ca hm s.

2.1 nh ngha. Gi s hm s f xc nh trn tp hp D .

a) Nu tn ti mt im 0x D sao cho ( ) ( )0f x f x vi mi x D th s

( )0M f x= c gi l gi tr ln nht ca hm s f trn D , k hiu l

( )maxx D

M f x

= .

b) Nu tn ti mt im 0x D sao cho ( ) ( )0f x f x vi mi x D th s

( )0m f x= c gi l gi tr nh nht ca hm s f trn D , k hiu l

( )minx D

m f x

= .

2.1 Nhn xt. Nh vy, mun chng t rng s M (hoc m ) l gi tr ln nht (hoc

gi tr nh nht) ca hm s f trn tp hp D cn ch r :

a) ( )f x M (hoc ( )f x m ) vi mi x D ;

b) Tn ti t nht mt im 0x D sao cho ( )0f x M= (hoc ( )0f x m= ).

2.2 Nhn xt. Ngi ta chng minh c rng hm s lin tc trn mt on th t

c gi tr nh nht v gi tr ln nht trn on .

Trong nhiu trng hp, c th tm gi tr ln nht v gi tr nh nht ca hm

s trn mt on m khng cn lp bng bin thin ca n.

I.2. Gi tr nh nht, gi tr ln nht ca hm s



t 2 2 2

( )f t + 0 2 2 ( )f t 2 2

Quy tc tm gi tr nh nht, gi tr ln nht ca hm f trn on ;a b nh sau :

1. Tm cc im 1 2, ,..., nx x x thuc khong ( );a b m ti f c o hm bng 0

hoc khng c o hm.

2. Tnh ( ) ( ) ( ) ( )1 2, ,..., ,nf x f x f x f a v ( )f b .

3. So snh cc gi tr tm c.

S ln nht trong cc gi tr l gi tr ln nht ca f trn on ;a b , s nh nht

trong cc gi tr l gi tr nh nht ca f trn on ;a b .

Trong mc ny chng ti trnh by mt s v d tm gi tr nh nht, gi tr ln

nht ca hm s.

Th d 1 ( thi tuyn sinh i hc khi B 2003)

Tm gi tr nh nht v gi tr ln nht ca hm s ( ) 24f x x x= +

Li gii. Tp xc nh 2;2D = , ( ) 21 4x

f xx

=

, ( ) 0 2f x x = =

Bng bin thin

T bng bin thin ta c ( ) ( )2;2

min 2 2x

f x f

= = v ( ) ( )2;2

max 2 2 2x

f x f

= = .

Th d 2. ( thi tuyn sinh i hc khi B 2004)

Tm gi tr ln nht v nh nht ca hm s 2ln x

yx

= trn on 31;e

Li gii. Ta c ( )

2

2 2

12 ln . . ln ln 2 lnx x x x xxy

x x

= =

T c bng bin thin :

x 1 2e 3e y 0 + 0

y 0

24e

39e

I.3. Mt s th d tm GTNN, GTLN ca hm s



Vy ( ) 232 24

1;max

eey y e x e

= = = v ( )31;

min 1 0 1ey y x

= = =

Tm gi tr nh nht v gi tr ln nht ca hm s

1) ( ) ( )22 231 2 1f x x x= +

2) ( ) 5cos cos5f x x x= vi 4 4x

3) ( )4 2 2

2 2

2 1 1 1 3

1 1 1

x x xf x

x x

+ + + +=

+ + +

Hng dn. t 2 21 1t x x= + + , vi 2 2t .

Bi tp tng t



CHNG II

K THUT GIM BIN TRONG BI TON TM GI TR NH NHT, GI TR LN NHT CA BIU THC

T kt qu ca Chng I chng ta thy rng vic tm GTNN, GTLN ca hm

s kh n gin. Vic chuyn bi ton tm GTNN, GTLN ca mt biu thc khng t

hn hai bin sang bi ton tm GTNN, GTLN ca hm s cha mt bin s gip chng

ta gi c bi ton tm GTNN, GTLN ca mt biu thc.

Trong phn ny chng ti trnh by mt s dng bi ton tm GTNN, GTLN ca biu

thc cha hai bin bng cch th mt bin qua bin cn li. T xt hm s v tm

gi tr nh nht, gi tr ln nht ca hm s.

Th d 1. Cho , 0x y > tha mn 54

x y+ = . Tm gi tr nh nht ca biu thc

4 14

Px y

= +

Li gii. T gi thit 54

x y+ = ta c 54

y x= . Khi 4 15 4

Px x

= +

.

Xt hm s ( ) 4 15 4

f xx x

= +

vi 50;4

x

. Ta c ( )( )2 2

4 4

5 4f x

x x = +

.

Bng bin thin T bng bin thin ta c ( ) ( )

50;4

min 1 5x

f x f

= = .

Do min 5P = t c khi 11,4

x y= = .

x 0 1 54

( )f x 0 + + +

( )f x 5

II.1. Tm GTNN, GTLN ca biu thc bng phng php th



Th d 2. Cho ,x y tha mn 20, 12y x x y + = + . Tm gi tr nh nht, gi tr ln nht ca biu thc 2 17P xy x y= + + + .

Li gii. T gi thit 20, 12y x x y + = + ta c 2 12y x x= + v 2 12 0x x+ hay 4 3x . Khi 3 23 9 7P x x x= + . Xt hm s ( ) 3 23 9 7, 4;3f x x x x x = + . Ta c ( ) ( )2' 3 2 3f x x x= + .

Ta c bng bin thin


min 1 12x

f x f

= = , ( ) ( ) ( )4;3

max 3 3 20x

f x f f

= = = .

Do min 12P = t c khi 1, 10x y= = v max 20P = t c khi

3, 6x y= = hoc 3, 0x u= = .

Th d 3. Cho , 0x y > tha mn 1x y+ = . Tm gi tr nh nht ca biu thc

1 1

x yP

x y= +

Li gii. T gi thit , 0x y > , 1x y+ = ta c 1 ,0 1y x x= < < .

Khi ta c 11

x xP

x x

= +

.

Xt hm s ( ) 11

x xf x

x x

= +

, ( )

( )2 1

2 1 1 2

x xf x

x x x x

+ =

.

Bng bin thin

T bng bin thin suy ra ( )

( )0;1

1min min 2

2xP f x f

= = = t c khi 1

2x y= = .

Nhn xt. Qua ba th d ny cho ta mt k thut gim bin khi tm GTNN, GTLN ca

biu thc hai bin bng cch th mt bin qua bin cn li v s dng cc gi thit

nh gi bin cn li. T tm GTNN, GTLN ca hm s cha mt bin b chn.

x 4 3 1 3 ( )f x + 0 0 +

20 20 ( )f x 13 12

x 0 12

1

( )f x 0 + + +

( )f x 2



1/ Cho , 3;2x y tha mn 3 3 2x y+ = . Tm gi tr nh nht, gi tr ln nht ca

biu thc 2 2P x y= + . 2/ Cho , 0x y tha mn 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu

thc 1 1x y

Py x

= ++ +

3/ Cho , 0x y > tha mn 1x y+ = . Tm gi tr nh nht ca biu thc

2 22 2

1 1P x y

x y= + + +

4/ Cho 1x y+ = . Tm gi tr nh nht ca biu thc

( ) ( )3 3 2 23 3P x y x y x y= + + + + 5/ Cho , , ,a b x y tha mn 0 , 4a b< , 7a b+ v 2 3x y .

Tm gi tr nh nht ca biu thc ( )

2 2

2 2

2 2x y x yP

xy a b

+ + +=

+

Hng dn. Tm gi tr ln nht ca 2 2Q a b= + l M , xt hm s

( ) ( )2 22 2

,.

x y x yg y f x y

xy M+ + +

= = vi n y v x l tham s, tm gi tr nh nht

ca ( )g y l ( )h x . Sau tm gi tr nh nht ca hm s ( )h x vi 2;3x . 6/ Cho ,x y tha mn 3x y . Tm gi tr nh nht ca biu thc

2 2 8 16P x y x= + + . Hng dn. Nu 0x > th 6 2x y t xt hm s ( ) 6 2 8 16f x x x x= + + . Nu

0x th 2 2 8 16 16x y x+ + vi mi 30,x x y . 7/ Cho ( ), 0;1x y tha mn 1x y+ = . Tm gi tr nh nht ca biu thc

x yP x y= + .

Hng dn. Xt hm s ( ) ( ), 0;1xf x x x= . Chng minh ( ) ( )2 2

f x f y x yf + +

.

Ta c ( ) ( ) ( )1 11 1 2 22

xxP x x f x f x f = + = + =

.

8/ Cho , 0x y > tha mn 2x y+ = . Chng minh rng x yxy x y .

Bi tp tng t



Trong phn ny chng ti trnh by mt s dng bi ton tm GTNN, GTLN ca biu

thc cha hai bin m gi thit hoc biu thc th hin tnh i xng. T bng

php t n ph ta chuyn v bi ton tm G ca hm s.

Th d 1. Cho 2 2x y x y+ = + . Tm gi tr nh nht, gi tr ln nht ca biu thc 3 3 2 2P x y x y xy= + + + Li gii.

t t x y= + , t gi thit 2 2x y x y+ = + ta c ( ) ( )2 22xy x y x y t t= + + =

hay 2

2t t

xy

= . p dng bt ng thc ( ) ( ) ( )2 2 22 2x y x y x y+ + = + hay 2 2t t suy ra 0 2t . Khi biu thc ( ) ( )3 22P x y xy x y t= + + = . Do ta

c max 4P = t c khi 2t = hay 2x y+ = v 1xy = suy ra 1x = v 1y = , ta c min 0P = t c khi 0t = hay 0x y= = .

Nhn xt. Bi ton ny gi thit v biu thc P c cho di dng i xng vi hai

bin. V vy, chng ta ngh n cch i bin t x y= + . Nhng gii bi ton trn

vn th phi tm iu kin ca bin t . Sau y l mt s bi ton vi nh hng tng

t.

Th d 2. Cho , 0x y > tha mn 2 2 1x xy y+ + = .

Tm gi tr ln nht ca biu thc 1

xyP

x y=

+ +

Li gii. t t x y= + . T gi thit , 0x y > v 2 2 1x xy y+ + = suy ra 2 1xy t= .

p dng bt ng thc ( )2 4x y xy+ suy ra 103

t< .

Khi 3 313

P t

= .

V vy 3 3max3

P

= t c khi ( ) 2 1; ;3 3

x y =

hoc ( ) 1 2; ;3 3

x y =

.

Th d 3. Cho ,x y tha mn 1x y+ v 2 2 1x y xy x y+ + = + + . Tm gi

tr nh nht, gi tr ln nht ca biu thc 1

xyP

x y=

+ +.

II.2. Tm GTNN, GTLN ca biu thc c tnh cht i xng



Li gii. t t x y= + . T gi thit 2 2 1x y xy x y+ + = + + ta c

( ) ( )2 1x y xy x y+ = + + hay 2 1xy t t= .

p dng bt ng thc ( )2 4x y xy+ suy ra 23 4 4 0t t hay 2 23t . Khi

2 1

1t t

Pt

=+

. Xt hm s ( )2 1

1t t

f tt

=+

, ( )( )

2

2

2

1

t tf t

t

+ =+

,

( ) 0f t = 0 2t t = = (loi). Bng bin thin T bng bin thin ta c ( ) ( )

23;2

min min 0 1t

P f t f

= = = t c khi

( ) ( ); 1;1x y = hoc ( ) ( ); 1; 1x y = v ( ) ( ) ( )23

23;2

1max max 2

3tP f t f f

= = = =

t c khi 13

x y= = hoc 1x y= = .

Th d 4. Cho ,x y tha mn 0 , 1x y< v 4x y xy+ = . Tm gi tr nh nht,

gi tr ln nht ca biu thc 2 2P x y xy= + .

t t x y= + . T gi thit 0 , 1x y< v 4x y xy+ = suy ra 4t

xy = v 1 2t .

Khi ( )2 2 334

P x y xy t t= + = . Xt hm s ( ) 2 34

f t t t= , ( ) 324

f t t = ,

( ) 0f t = 38

t = (loi). Bng bin thin


1min min 1

4tP f t f

= = = t c khi 1

2x y= = v

( ) ( )1;2

5max max 2

2tP f t f

= = = t c khi ( ) ( )2 22 22 2; ;x y += hoc

( ) ( )2 2 2 22 2; ;x y + = .

t 23 0 2 ( )f t 0 +

( )f t 13

1

13

t 1 2 ( )f t +

( )f t 14

52



Th d 5. Cho ,x y tha mn , 0x y v ( ) 2 2 2xy x y x y x y+ = + + . Tm

gi tr ln nht ca biu thc 1 1Px y

= + .

Li gii. t t x y= + . T gi thit ( ) 2 2 2xy x y x y x y+ = + + hay

( ) ( ) ( )2 2 2xy x y x y xy x y+ = + + + suy ra 2 2

2t t

xyt +

=+

. p dng bt ng

thc ( )2 4x y xy+ suy ra 3 22 4 8

02

t t tt

+

+ hay 2 2t t< . Khi

2

2

22

x y t tP

xy t t

+ += =

+. Xt hm s ( )

2

2

22

t tf t

t t

+=

+, ( )

( )

2

22

3 4 4

2

t tf t

t t

+ + = +

,

( ) 0f t = 232t t = = (loi). Bng bin thin T bng bin thin ta c ( ) ( )

2 2max max 2 2

t tP f t f

tha mn 3xy x y+ + = .

Chng minh rng 2 23 3 31 1 2x y xy

x yy x x y

+ + + ++ + +

Li gii. t t x y= + . T gi thit , 0x y > , 3xy x y+ + = v p dng bt ng

thc ( )2 4x y xy+ ta c 3 , 0xy t t= > v 2 4 12 0t t+ hay 2t hoc 6t (loi). Khi bt ng thc cn chng minh tr thnh

( ) ( )( ) ( )

223 6 3 32

1 2

x y xy x y xyx y xy

xy x y x y

+ + ++ + +

+ + + + hay

22 3 23 9 18 3 92 4 12

4 2t t t

t t t t tt

+ + + + ( )( )22 6 0t t t + +

lun ng vi mi 2t , du bng xy ra khi 2t = hay 1x y= = .

Th d 7. Cho , 0x y > tha mn 2 2 1x y+ = . Tm gi tr nh nht ca biu thc

( ) ( )1 11 1 1 1P x yy x

= + + + + + .

Li gii. t t x y= + . T gi thit , 0x y > v 2 2 1x y+ = suy ra 2 12t

xy

= v

1t > . p dng bt ng thc ( ) ( )2 2 22x y x y+ + suy ra 1 2t< .

t 2 23 2 +

( )f t 0 + 0 _

( )f t 1

0

2

1



Khi ( )2

11

x y t tP x y xy

xy t

+ + = + + + = . Xt hm s ( )

2

1t t

f tt+

=

,

( )( )

2

2

2 1

1

t tf t

t

=

, ( ) 0 1 2f t t = = (loi). Bng bin thin

T bng bin thin ta c

(( ) ( )

1; 2min min 2 4 3 2

tP f t f

= = = + t c khi

1

2x y= = .

Nhn xt. Qua cc th d trn, cho ta mt k thut gim bin ca bi ton tm GTNN,

GTLN ca biu thc hai bin c tnh i xng: Do tnh i xng nn ta lun c th

bin i a v mt trong cc dng t t x y= + , 2 2t x y= + hoc t xy= , t a

v tm GTNN, GTLN ca hm s.

1/ Cho , 0x y > tha mn 1 3x y xy+ + = . Tm gi tr ln nht ca biu thc

( ) ( ) 2 23 3 1 11 1

x yP

y x x y x y= +

+ +

2/ Cho ,x y khng ng thi bng 0 v tha mn 1x y+ = . Tm gi tr nh nht ca

biu thc 2 2

2 2 2 2

11 1

x yP

x y y x= + +

+ + +

3/ Cho 2 2 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu thc

1 1P x y y x= + + + 4/ Cho 2 2 1x y+ = . Tm gi tr nh nht, gi tr ln nht ca biu thc

1 1

x yP

y x= +

+ +

5/ Cho , 0x y thay i tha mn ( ) 2 2x y xy x y xy+ = + . Tm gi tr ln nht ca

biu thc 3 3

1 1P

x y= + .

6/ Cho ,x y tha mn 2 2 2x xy y+ + . Tm gi tr ln nht ca biu thc 2 2P x xy y= + .

t 1 2

( )f t

( )f t

+

4 3 2+

Bi tp tng t



Trong phn ny chng ti trnh by mt s dng bi ton tm gi tr nh nht, gi tr ln nht ca biu thc cha hai bin m gi thit hoc biu thc th hin tnh ng cp. T xt hm s v tm gi tr nh nht, gi tr ln nht ca hm s.

Th d 1. Cho , 0x y > tha mn 2 2 1x y+ = . Tm gi tr ln nht ca biu thc

( )P y x y= + .

Li gii. t y tx= . T iu kin , 0x y > suy ra 0t > . T gi thit 2 2 1x y+ = ta

c 22

11

xt

=+

. Khi biu thc ( )2

22

11

t tP x t t

t

+= + =

+. Xt hm s ( )

2

2 1t t

f tt

+=

+,

( )( )

2

22

2 1

1

t tf t

t

+ + =+

, ( ) 0 2 1 1 2f t t t = = + = (loi). Bng bin thin

T bng bin thin ta c ( ) ( ) 2 120max max 2 1tP f t f

+

>= = + = t c khi

( ) ( )2 12 12 2 2 2; ;x y += .

Th d 2. Cho , 0x y v tha mn 2 2 1x y+ = . Tm gi tr ln nht ca biu thc 2

2

4 6 52 2 1x xy

Pxy y

+ =

Li gii.

Nu 0x = th t gi thit 2 2 1x y+ = v 0y suy ra 1y = . Khi 53

P = .

Nu 0x th t y tx= . T gi thit , 0x y v 2 2 1x y+ = suy ra 0t v

22

11

xt

=+

. Khi ( )( )2 2

2 2 2

4 6 5 5 6 12 2 1 3 2 1

x t t tP

x t t t t

+ += =

+. Xt hm s

( )2

2

5 6 13 2 1t t

f tt t

+=

+, ( )

( )

2

22

8 4 4

3 2 1

t tf t

t t

+ = +

, ( ) 10 12

f t t t = = = (loi).

Bng bin thin

t 0 2 1+ +

( )f t + 0

( )f t

0

2 12+

1

II.3. Tm GTNN, GTLN ca biu thc c tnh ng cp



T bng bin thin ta c ( ) 5 , 03

f t t< v ( ) ( )120min min 1tP f t f= = = t

c khi 25

x = v 15

y = . V vy 5max3

P = t c khi ( ) ( ); 0;1x y = v

min 1P = t c khi ( ) ( )2 15 5; ;x y = .

Th d 2. Cho , 0x y > . Chng minh rng ( )

2

32 2

4 184

xy

x x y

+ +

Li gii. t xty

= . T gi thit , 0x y > suy ra 0t > . Khi bt ng thc cn

chng minh tng ng vi ( )32

4 184

t

t t

+ + hay ( )32 4 2t t t+ . Xt hm

s ( ) ( )32 4f t t t t= + ,

( ) ( ) ( ) ( ) ( )3 3

2 2 23

2

2 2

3 4 4 4 34

4 4

t t t t t t tf t t t

t t

+ + + = + =

+ +,

( ) 220f t t = = . Ta c bng bin thin

T bng bin thin ta c ( ) ( )220max 2t f t f> = = hay ( )3

2 4 2t t t+ du bng

xy ra khi 22

t = hay 2y x= .

t 0 22

+ ( )f t + 0

( )f t 0

2

0

t 0 12 +

( )f t 0 +

( )f t

1 1

53



1/ Cho , 0x y > tha mn 1xy y . Tm gi tr nh nht ca biu thc

2 3

2 39

x yP

y x= +

2/ Cho , 0x y . Chng minh rng 3 3 23 7 9x y xy+ .

3/ Cho , 0x y . Chng minh rng 4 4 3 3x y x y xy+ + .

4/ Tm gi tr nh nht ca biu thc 2 2

2 23 8x y x y

Py xy x

= + + vi , 0x y .

Trong phn ny chng ti trnh by mt s dng bi ton tm gi tr nh nht, gi tr

ln nht ca biu thc cha ba bin bng cch t n ph hoc th hai bin qua mt

bin cn li. T , chuyn c bi ton v bi ton tm gi tr nh nht, gi tr ln

nht ca hm s.

Th d 1. Cho , , 0x y z > tha mn 1x y z+ + = . Chng minh rng 1 1 16xz yz

+

Li gii. t t x y= + . T gi thit ta c ( )1 1z x y t= + = v 0 1t< < .

p dng bt ng thc ( )2 4x y xy+ hay 2

4t

xy .

Khi ( ) 2

1 1 41t

Pxz yz xy t t t

= + = +

.

Xt hm s ( ) 24

f tt t

= +

, ( ) ( )( )224 2 1t

f tt t

=

+, ( ) 10

2f t t = = .

Ta c bng bin thin

t 0 12 1

( )f t 0 +

( )f t

+

16

+

Bi tp tng t

II.4. Tm GTNN, GTLN ca biu thc ba bin



T bng bin thin ta c ( )

( ) ( )120;1min 16t f t f = = t c khi 1 14 2,x y z= = = .

V vy 1 1 16xz yz

+ .

Th d 2. Cho 2 2 2 1x y z+ + = . Tm gi tr nh nht, gi tr ln nht ca biu thc P x y z xy yz zx= + + + + +

Li gii. t t x y z= + + . p dng bt ng thc Cauchy Schwarz ta c

( ) ( )2 2 2 23x y z x y z+ + + + suy ra 3 3t . Khi

( ) ( ) ( ) ( )2 2 2 2 21 1 2 12 2

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

Xt hm s ( ) ( )21 2 12

f t t t= + , ( ) 2 2f t t = + , ( ) 0 1f t t = = . Ta c bng bin thin T bng bin thin ta c ( ) ( )

3; 3min min 1 1

tP f t f

= = = t c khi 1t =

hay ( ) ( ); ; 1;0;0x y z = v cc hon v ca n; ( ) ( )

3; 3max max 3 1 3

tP f t f

= = = + t c khi 3t = hay

( ) ( )1 1 13 3 3; ; ; ;x y z = .

Th d 3. Cho , , 0x y z tha mn 1x y z+ + = .

Chng minh rng 3 3 3 15 14 4

x y z xyz+ + +

Li gii. Do vai tr ca , ,x y z bnh ng nn ta lun gi s c { }min , ,x x y z= . T

gi thit , , 0x y z , 1x y z+ + = ta c 103

x v 1y z x+ = . p dng bt

ng thc ( )2

4

y zyz

+ v 27 3 0

4x < . Khi biu thc

( ) ( )33 3 3 315 1534 4

P x y z xyz x y z yz y z xyz= + + + = + + + +

( ) ( ) ( )3 33 315 273 1 34 4x x

x y z yz y z x x yz = + + + + = + +

t 3 1 3

( )f t 0 +

( )f t 1 3

1

1 3+



( ) ( ) ( )2

33 3 227 11 3 27 18 3 44 4 16

y z xx x x x x

+ + + = + +

Xt hm s ( ) ( )3 21 27 18 3 416

f x x x x= + + , ( ) ( )21 81 36 316

f x x x = + ,

( ) 109

f x x = =13

x = . Bng bin thin

T bng bin thin ta c ( ) ( ) ( )13

130;

1max 0

4xf x f f

= = = . Do 14

P . Du bng xy

ra khi ( ) ( )1 1 13 3 3; ; ; ;x y z = hoc ( ) ( )1 12 2; ; 0; ;x y z = v cc hon v ca n.

Th d 4. Cho ( ), , 0;1x y z tha mn 1xy yz zx+ + = .

Tm gi tr nh nht ca biu thc 2 2 21 1 1

x y zP

x y z= + +

.

Li gii. Ta c ( ) ( ) ( )

2 2 2

2 2 21 1 1x y z

Px x y y z z

= + +

. Xt hm s ( ) ( )211

f tt t

=

vi 0 1t< < , ( )( )

2

22 2

3 1

1

tf t

t t

=

, ( ) 13

10

3f t t t = = = (loi).

Ta c Bng bin thin

T bng bin thin ta c ( ) ( )21 3 3

0;121t

t t

.

V vy ( ) ( )2 2 23 3 3 3 3 32 2 2

P x y z xy yz zx + + + + =

Do 3 3min2

P = t c khi 13

x y z= = = .

x 0 19

13

( )f x + 0 0

( )f x 14

727

14

t 0 13

1

( )f t 0 +

( )f t +

3 32

+



Chng III

MT S BI TON TRONG CC THI I HC

Bi 1 ( thi tuyn sinh i hc A 2011)

Cho , ,x y z l ba s thc thuc on 1;4 v ,x y x z . Tm gi tr nh nht ca

biu thc 2 3x y z

Px y y z z x

= + ++ + +

Li gii. Trc ht ta chng minh : 1 1 21 1 1a b ab

+ + + +

(*), vi a v b dng

v 1ab . Tht vy, ( ) ( )( ) ( )( )* 2 1 2 1 1a b ab a b + + + + +

( ) 2 2a b ab ab a b ab + + + + ( )( )21 0ab a b , lun ng vi ,a b dng v 1ab . Du bng xy ra, khi v ch khi : a b= hoc 1ab = .

p dng (*), vi x v y thuc on 1;4 v x y , ta c:

1 1 1 22 3 3

1 1 2 1

xP

x y z x y xy z x y

= + + ++

+ + + +

Du bng xy ra, khi v ch khi : z xy z= hoc 1x

y= (1)

t , 1;2x t ty

= . Khi : 2

2

212 3

tP

tt +

++. Xt hm s :

( )2

2

2, 1;2

12 3t

f t ttt

= + ++, ( )

( ) ( )

( ) ( )

3

2 22

2 4 3 3 2 1 90

2 3 1

t t t tf t

t t

+ + = , suy ra : )( )

52;

5 23min

2 4f t f

+

= = .

Vy, 23min4

P = t khi v ch khi : 52

a bb a+ =

v 1 12a ba b

+ = + ( ) ( ); 2;1a b = hoc ( ) ( ); 1;2a b =

Bi 3 ( thi tuyn sinh i hc khi B-2010)

Cho cc s thc khng m , ,a b c thon mn 1a b c+ + = . Tm gi tr nh nht ca

biu thc ( ) ( )2 2 2 2 2 2 2 2 23 3 2M a b b c c a ab bc ca a b c= + + + + + + + +

Li gii. Ta c: ( ) ( ) ( )2 3 2 1 2M ab bc ca ab bc ca ab bc ca + + + + + + + +

t t ab bc ca= + + , ta c : ( )2 1

03 3

a b ct

+ + = .

Xt hm s ( ) 2 3 2 1 2f t t t t= + + trn 10;2

, ta c : ( ) 22 3

1 2f t t

t = +



( )( )32

2 01 2

f tt

=

, du bng ch xy ra ti 0t = , suy ra ( )f t nghch bin.

Xt trn on 10;3

ta c : ( ) 1 11 2 3 03 3

f t f = >

, suy ra ( )f t ng bin. Do

: ( ) ( ) 10 2, 0;3

f t f t =

.

V th : ( ) 12, 0;3

M f t t

. 2 , 1M ab bc ca ab bc ca= = = + + = v

1a b c+ + = ( ); ;a b c l mt trong cc b s : ( ) ( ) ( )1;0;0 , 0;1;0 , 0;0;1

Do gi tr nh nht ca M l 2.

Bi 4. ( thi tuyn sinh i hc khi B 2009)

Tm gi tr nh nht ca biu thc ( ) ( )4 4 2 2 2 23 2 1A x y x y x y= + + + + vi ,x y l

cc s tha mn ( )3 4 2x y xy+ + .

Li gii. Da vo bt ng thc hin nhin : ( )2 4x y xy+ nn

( ) ( ) ( ) ( )3 3 2 34 2 4 2x y xy x y x y x y xy+ + + + + + +

( ) ( )3 2 2 0x y x y + + + ( ) ( ) ( )21 2 0x y x y x y + + + + + (1)

Do ( ) ( ) ( )2

2 1 72 0

2 4x y x y x y

+ + + + = + + + >

v t (1) suy ra : 1x y+ .

Vy nu cp ( );x y tha mn yu cu bi th 1x y+ (2).

Ta bin i A nh sau:

( ) ( )4 4 2 2 2 23 2 1A x y x y x y= + + + +

( ) ( ) ( )22 2 4 4 2 23 3 2 12 2x y x y x y= + + + + + (3)

Do ( )22 24 4

2

x yx y

++ nn t (3) suy ra :

( ) ( ) ( ) ( ) ( )2 2 22 2 2 2 2 2 2 2 2 23 3 92 1 2 12 4 4

A x y x y x y x y x y + + + + + = + + +

V ( )22 22

x yx y

++ nn t (2) ta c : 2 2 1

2x y+ .



t ( ) 29 2 14

f t t t= + vi 2 2 12

t x y= + . Ta c : ( ) 9 12 0,2 2

f t t t = > .

Suy ra : ( )12

1 9min

2 16tf t f

= = (4)

T (4) suy ra 916

A . Mt khc d thy khi 12

x y= = th 916

A = .

Vy 9min16

A = khi 12

x y= = .

Bi 5. ( thi tuyn sinh i hc khi D 2009)

Cho , 0x y v 1x y+ = . Tm gi tr ln nht v nh nht ca biu thc :

( )( )2 24 3 4 3 25S x y y x xy= + + +

Li gii. Ta c : ( )( ) ( )2 2 2 2 3 34 3 4 3 25 16 12 34S x y y x xy x y x y xy= + + + = + + +

( )( )2 2 2 216 12 34x y x y x xy y xy= + + + + ( )22 216 12 3 34x y x y xy xy = + + + 2 216 2 12x y xy= + (1) (do 1x y+ = ).

t xy t= . Vi 0, 0x y , ta c : ( )2 1 1

0 04 4 4

x yxy t

+ =

Xt hm s ( ) 216 2 12f t t t= + vi 104

t . Ta c : ( ) 32 2f t t = . Bng bin thin

Suy ra ( ) ( )14

1160;

191min

16f t f

= = v ( ) ( )14

140;

25max

2f t f

= = .

Vy: Gi tr nh nht ca S t c

2 3 2 31 ;1 4 4116 2 3 2 3;16 4 4

x y x yt

xyx y

+ + = = = = += = =

Gi tr ln nht ca S t c 1

1 114 24

x yt x y

xy

+ = = = = =

t 0 116

14

( )f t 0 +

( )f t 12

19116

252



Sng kin ny t c mt s kt qu sau :

+ Nhc li cch tm GTNN, GTLN ca hm s thng qua mt vi v d.

+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha hai bin

bng cch th mt bin qua bin cn li.


bng cch t n ph theo tnh i xng t x y= + , 2 2t x y= + hoc t xy= .


bng cch t n ph theo tnh ng cp xty

= ..

+ H thng mt s dng bi ton tm GTNN, GTLN ca mt biu thc cha ba bin

bng cch t n ph hoc th hai bin qua mt bin cn li.

Qua thc t ging dy chng ti thy rng vn no d kh m gio vin quan tm

v truyn th cho hc sinh bng lng say m v nhit tnh ca mnh th s cun ht cc

em vo con ng nghin cu. Bi ton tm GTNN, GTLN ca mt biu thc khng

phi l mt vn mi, nhng thc t cho thy cn nhiu Thy, C cha quan tm

ng mc vn ny.

Vi sng kin kinh nghim ny hy vng gp thm mt ti liu cho qu Thy, C v

cc bn ng nghip ; gip cc em hc sinh c thm nhng kinh nghim cho loi ton

ny, t t tin hn khi thi i hc.

Phn kt lun

2. Bi hc kinh nghim

3. ngha ca SKKN

1. Kt qu t c



Sng kin kinh nghim ny c th trin khai nh mt chuyn bi dng hc sinh

gii ; cng nh dng ging dy cho cc em hc sinh n tp thi i hc, nhm gip

cc em hc sinh c th vt qua tr ngi tm l t trc ti nay cho loi bi ton ny.

1. Sch Gio khoa Gii tch 12, Nh xut bn Gio dc Vit Nam.

2. Tuyn tp Tp ch Ton hc v tui tr nm 2008, 2009, 2010 ; v Tp ch Ton hc

v tui tr hng thng.

3. Ngun Internet : http://www.VnMath.com.vn,

4. Kh nng ng dng v trin khai

Ti liu tham kho

skkn_kythuatgiambientimgtnngtln

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