phuong phap chuan hoa

7/23/2019 Phuong Phap Chuan Hoa

1/5

PPHHNNGGPPHHPPCCHHUUNNHHOO

1. t vn :Cho H(x, y, z) l mt a thc ng cp bc k, ngha lH(tx, ty, tz) = t

kH(x, y, z)

v h/s F(x, y, z) tha mn F(x, y, z) = F( x, y, z). Khi gi trcaF(x, y, z) trn min {(x, y, z)/H(x, y, z) = a, a > 0}khng thay i khi a thay i.

Tht vy, gisM(x, y, z): H (x, y, z) = a1

M(x, y, z): H(x, y, z) = a2; a1a2; a1, a2> 0

Ta c ( ) 21 21

aH x, y, z = a H(x, y, z) = a

a

k

2 2 2 2k k k k

2 2

1 1 1 1

a a a aH(x, y, z) = a H x, y, z = a

a a a a

t 2 2 2k k k1 1 1

a a ax' = x, y' = y, z' = z

a a a

Ta c: 2H(x', y', z') = a F(x', y', z') = F(x, y, z)

Mt khc : { } { } 1 2M H(x, y, z) = a M' H(x', y', z') = a

Nhvy tm gi trca F(x, y, z) trn min H(x, y, z) ta chcn tm gitrca F(x, y, z) trn min H(x, y, z) = a cnh thch hp.2. Cc bi ton p dng.

Bi ton1: Cho a, b, c> 0. Tm max

2 2 2 2 2 2

a(b + c) b(c + a) c(a + b)Q = F(a, b, c) = + +

(b + c) + a (c + a) + b (a + b) + c

( Olimpic 30 - 4- 2006).

Li gii:Do F(a, b, c) = F(ta, tb, tc) nn ta tm gi trQ trn min a + b + c = 1Ta c:

2 2 2

a(1- a) b(1- b) c(1- c)Q = + +

1 - 2a + 2a 1- 2b + 2b 1- 2c + 2c

Theo Csi:

>

2 2

22

2a + 1- a (a + 1)2a(1- a) =

2 4

(a+1) (1- a)(a + 3)1- 2a + 2a = 1- 2a(1- a) 1 - = 0

4 4


2/5

2

a(1- a) 4a(1- a) 4 3= = 4 1 -

(1- a)(a + 3) a + 3 a + 31- 2a + 2a

3 3 3 1 1 1Q 4 1- + 1- + 1 - = 4 3 - 3 + +

a + 3 b + 3 c + 3 a + 3 b + 3 c + 3

Ta c:

1 1 1 9 9 9 6

+ + = Q 4(3 - 3. ) =a + 3 b + 3 c + 3 a + b + c + 9 10 10 5

Suy ra :6

maxQ =5

khi a = b = c

Bi ton 2:Cho a, b, c > 0. Tim min

2 3 3 3 2 2 2

2 2 2

(a + b + c) 1 a + b + c a + b + cQ = + - (1)

2 abc ab + bc + caa + b + c

Li gii: Do F(a, b, c) = F(ta, tb, tc). Ta chtm gi trca Q trn min

a2 + b2 + c2 = 3

Khi :

[ ]

[ ]

2 2 2 2 2

3 3 3

3 3 3

(a + b + c) = a + b + c + 2ab + 2bc + 2ca (a + b + c) = 3 + 2(ab + bc + ca)

a + b + c = 3abc + (a + b + c) 3 - (ab + bc + ca)

a + b + c 1 1 1= 3 + ( + + ) 3 - (ab + bc + ca)

abc ab bc ca

t

1 1 1 9

= ab + bc + ca 3; = + +ab bc ca

Suy ra:

5 2 9 3 2 12 6(3 ) 2 2 2( )

2 3 2 2 3 3Q

+ + = + + = + +

13

1 / 36 3 3 3 3

3( ) 2 2 43 3 Q

+ = + +

+

Suy ra: minQ = 4 , khi a = b = c > 0

Bi ton 3:Cho a, b, c > 0. Chng minh

37(a + b + c)(ab + bc + ca) 9abc + 2(a + b + c) (1)


3/5

Li gii: 2 3

7(ab + bc + ca) 9abc(1) F(a, b, c) = - 2

(a + b + c) (a + b + c)

Do F(a , b, c) = F(ta, tb, tc). Ta c thxem a + b + c = 1.Suy ra: F(a, b, c) = 7(ab + bc + ca) - 9abc = 7a(1 - a) + bc(7 - 9a)

Gis: 0 < a b c Ta c:

2 2a+b+c=1 1 (b + c) (1- a)0 < a ; 7 - 9a > 0; bc =

3 4 40 < a b c

Khi :

2

3 2

(1- a) 1F(a, b, c) 7a(1- a) + (7- 9a); 0 < a

4 3

1 1F(a, b, c) f(a) = (- 9a - 3a + 5a + 7)

4 4

Kho st hm sf(a), ta c: F(a, b, c) 2; F(a, b, c) = 2 a = b = c

Ch :Bt ng thc (1) c dng f(a, b, c) g(a, b, c) , trong f(a, b, c)v g(a, b, c) ng bc

f(x, y, z) v g(x, y, z) ng bc m (nguyn dng) nu

m

m

f( x, y, z) = f(x, y, z)

g( x, y, z) = g(x, y, z)

Cho bt ng thc: f(x, y, z) g(x, y, z) (*)Vi f, g ng bc v H(x, y, z) l mt a thc ng cp bc k. Nu (*) ng

trn min H(x, y, z) = a1th cng ng trn min H(x,, y,, z,) = a2vi a1,

a2 > 0. Tht vy:

2 2 2k k k1 2

1 1 1

m

2k

1

a a aH(x, y, z) = a H(x',y',z') = a ; x' = x; y; z

a a a

af(x', y', z') = .f(x, y, z)

a

Tng t:

m

2k

1

ag(x', y', z') = .g(x, y, z)

a

Khi : f(x, y, z) g(x, y, z) f(x', y', z') g(x', y', z')


4/5

Vy chng minh (*)ng trn min H(x,y,z) chcn chng (*) ng trnmin H(x, y, z) = a > 0 cnh. Vic chn gi tra l rt quan trng.Bi ton 4: Cho a, b, c l cc sthc. Chng minh

32 2 2 2 2 2 26(a + b + c)(a + b + c ) 27abc + 10(a + b + c ) (1)

(Olimpic Vit Nam 2004 )

Li gii: BT ng khi a = b = c = 0Nu 2 2 2a + b + c > 0 Chun ha 2 2 2a + b + c = 9

(1) 2(a + b + c) - abc 10

Gis:

2 2 2

2 b + c 9 - aa b c a 3 bc = 32 2

[ ]

22

2 2 2

VT = a(2 - bc) + 2(b + c) VT = a(2 - bc) + 2(b + c)

a + (b + c) (2 - bc) + 4

t : [ ] t = bc t -3; 3

[ ] 2 2VT = (9 + 2t) (2 - t) + 4 = f(t); t -3;3

f(t) f(t) 100 VT 10 (pcm)Bi ton 5:Cho a, b, c l ba cnh ca mt tam gic. Chng minh rng

(b + c- a)(a + c - b) + (a + b - c)(a + c - b) + (a + b - c)(b + c - a)

abc( a+ b + c) (1)

Li gii: t a = x2, b = y2, c = z2.

4 4 4 2 2 2 2 2 2(1) x + y + z + xyz(x + y + z) 2(x y + y z + z x ) (2)

Chun ha: 1y z+ + = . Ta c: (2) 1 + 9xyz 4(xy + yz + zx) 4(xy + yz + zx) - 9xyz 1 (*)

Gis: 1

0 < x y z 0 < x3

. Khi :

1

VT = 4x(1- x) + yz(4 - 9x) f(x)4

; 3 2f(x) = - 9x + 6x - x + 4

Kho st: 3 2f(x) = - 9x + 6x - x + 4 trn 10;3

f(x) 4 VT 1 . Suy ra (*) c chng minh


5/5

Bi tp tng t:

3

2 2 2

11. (a + b)(b + c)(c + a) + abc (a + b + c) ; a, b, c > 0

3

a + b + c 8abc2. + 2; a, b, c>0ab + bc + ca (a + b)(b + c)(c + a)

1 1 1 4abc3. (a + b+ c)( + + ) + 5; a

a + b b + c c + a (a + b)(b + c)(c + a), b, c > 0

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