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V MT CCH TM GI TR LN NHT , NH NHTCA BIU THC CHA HAI BIN S

B Ch - Thi Bnh tng www.mathvn.com

Bi vit ny chng ti xin trao i v mt phng php tm gi tr ln nht (GTLN), gi trnh nht (GTNN) ca biu thc cha hai bin s nh tp gi tr, trong hai bin b rngbuc bi mt iu kin cho trc .Bi ton : Cho cc s thc x , y tho mn iu kin : G(x ; y) = 0 .

Tm GTLN , GTNN (nu c) ca biu thc P = F(x ; y).Phng php gii :Gi T l tp gi tr ca P. Khi , m l mt gi tr ca T khi v ch khi h sau c nghim (x;y):

G(x;y) 0F(x;y) m

Sau tm cc gi tr ca tham s m h trn c nghim . T suy ra tp gi tr T ca P ,ri suy ra GTLN , GTNN (nu c) ca P.Sau y l cc bi ton minh ho .

Bi ton 1 : Cho hai s thc x , y tho mn iu kin : 3 3 3 3 3x( x 1) y y 1 xy Tm GTLN , GTNN ca biu thc 3 3 3F x y xy .

Li gii : Gi T1 l tp gi tr ca F . Ta c 1m T h sau c nghim: 3 3 3 3 33 3 3

x( x 1) y y 1 xyx y xy m

(I)

t3 3

3

S x yP xy

th x, y S,P : 2S 4P

H (I) tr thnh2 2S S 3P 0 S 2S 3m 0 (II)S P m P m S

Ta c :

22 2 24(S S)S 4P S S 4S 0 0 S 43

T , h (I) c nghim h (II) c nghim ( S ; P ) tho mn 2S 4P phng trnh

2S 2S 3m 0 c nghim S : 0 S 4 , iu ny xy ra khi v ch khiS

1

2

1 3m 0 1m

30 S 1 1 3m 41 1 3m 50 S 1 1 3m 4

0 m 8 . Do : 1T 0;8

Vy : minF = 0 , maxF = 8.Bi ton 2 : Cho cc s thc x, y tho mn : 32 2x - xy + y

Tm GTLN , GTNN ca biu thc 2 2G = x + xy - 2yLi gii : Gi T2 l tp gi tr ca G . Ta c 2m T h sau c nghim:

3 2 2

2 2x - xy + yx + xy - 2y = m (III)

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Nu y = 0 th h (III) tr thnh 2

23x

x m

33

xm

Nu y 0 th t x = ty ta c h :2

2 2 2

2 2 2

2

3yy (t t 1) 3 t t 1y (t t 2) m 3(t t 2) mt t 1

2

2

3y t t 1(m 3)t (m 3)t m 6 0

(IV)

Trng hp ny h (III) c nghim h (IV) c nghim y 0 phng trnh2(m 3)t (m 3)t m 6 0 (2) c nghim :

Nu m = 3 th (2) c nghim t = 32

Nu m 3 th (2) c nghim 2t 3m 6m 81 0 1 2 7 m 1 2 7 (m 3 )

Kt hp cc trng hp trn ta c cc gi tr ca m h (III) c nghim l :1 2 7 m 1 2 7 . Do : 2T 1 2 7 ; 1 2 7

Vy : minG = 1 2 7 , maxG = 1 2 7 Bi ton 3 : (Tuyn sinh i hc khi A nm 2006 )

Cho hai s thc thay i x 0 , y 0 tho mn : 2 2(x y)xy x y xy

Tm gi tr ln nht ca biu thc 3 31 1Ax y

Li gii : Gi T3 l tp gi tr ca A . Ta c 3m T h sau c nghim x 0 , y 0 :2 2 2 22 2

2 2 2

3 3 3 3

(x y)xy x y xy (x y)xy x y xy(x y)xy x y xy1 1 (x y)(x y xy) xy(x y)

m m mx y (xy) (xy)

2

2

(x y)xy (x y) 3xyx y( ) mxy

(V)

tS x yP xy

( 2S 4P ) , ta c h :2

2

SP S 3PS( ) mP

(VI)

H (V) c nghim x 0 , y 0 h (VI) c nghim ( S ; P ) tho mn 2S 4P .V 2 2 2 21 3SP x y xy (x y) y 0

2 4 vi mi x 0 , y 0 S 0

P vi mi x 0 ,

y 0T :

Nu m 0 th h (V) v nghim Nu m > 0 th t phng trnh 2S S( ) m m

P P S m.P thay vo phng

trnhu ca h (VI) c : 2 2mP mP 3P (m m)P 3 ( v SP > 0 nn P 0 )

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c P t phng trnh ny th m m 0 m 1 ( m > 0 ) v ta c3P

m( m 1) , do

3Sm 1

. Trng hp ny h (VI) c nghim ( S ; P )tho

mn 2S 4P khi v ch khi :

23 12( )m 1 m( m 1)

24( m 1)3 3 m 4( m 1) m 4

m( m 1)

0 m 16 (m 1) Tm li cc gi tr ca m h (V) c nghim x 0 , y 0 l : 0 m 16 , m 1 Do : 3T 0;16 \ 1Vy : maxA = 16 ( ch khng tn ti minA )

Bi ton 4 : ( HSG quc gia - Bng A + B nm 2005 )Cho hai s thc x, y tho mn : x 3 x 1 3 y 2 y Hy tm gi tr ln nht v nh nht ca biu thc K x y

Li gii : KX : x 1, y 2 Gi T4 l tp gi tr ca K . Ta c 4m T h sau c nghim:

3( x 1 y 2) mx 3 x 1 3 y 2 y (VII)x y m x y m

t u x 1 v v y 2 th u, v 0 v h (VII) tr thnh :

2 2 2

mu v3(u v) m 3

u v m 3 1 muv ( m 3)

2 9

u , v l hai nghim ca phng trnh :

22 2 2m 1 mt t ( m 3) 0 18t 6mt m 9m 27 0

3 2 9 (3)

T , h (VII) c nghim ( x ; y ) sao cho x 1, y 2 khi v ch khi (3) c hai nghimkhng m v iu kin l :

2t

t

2

t

9(m 18m 54) 0m 9 3 21S 0 m 9 3 153 2m 9m 27P 0

18

. Do 49 3 21T ;9 3 15

2

Vy : minK = 9 3 212

, maxK = 9 3 15

Bnh lun: u th ca phng php trn l quy bi ton tm GTLN, GTNN v bi ton tmtham s h c nghim , v vy khng cn ch r gi tr ca bin s biu thc t GTLN,GTNN . Nu dng cc bt ng thc nh gi th nht thit phi ch r cc gi tr ca bins ti biu thc t GTLN, GTNN. Cc bn c th m rng phng php ny cho biuthc c nhiu hn hai bin s .Cui cng l cc bi tp minh ho phng php trn :Bi 1 : Cho hai s thc x , y tho mn : 2 2 2( ) 7x y x y .

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Tm gi tr ln nht , nh nht ca biu thc 3 3P x(x 2) y(y 2)Bi 2 : Cho cc s thc x, y tho mn : 2 24x - 3xy +3y = 6 .

Tm gi tr ln nht v nh nht ca biu thc 2 2F x xy 2y Bi 3 : Cho cc s thc khng m x , y tho mn : x y 4 .

Tm gi tr ln nht v nh nht ca biu thc 1 9Q x y Bi 4 : Cho cc s dng x , y tho mn : xy x y 3

Tm GTLN ca biu thc 2 23x 3yG x yy 1 x 1

Bi 5 : (Cao ng kinh t k thut nm 2008) .Cho hai s x , y tho m n : 2 2x y 2

Tm GTLN , GTNN ca biu thc 3 3P 2(x y ) 3xy Bi 6 : (i hc Khi B nm 2008). Cho hai s thc x , y thay i v tho mn hthc: 2 2x y 1

Tm GTLN , GTNN ca biu thc2

2

2(x 6xy)P1 2xy 2y

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