. x^, 4x +3 y, Ta thay phuong trinh c6 nghỉm khi va chi khi — va
a) Dieu ki|n :x > 1 Xet phuang trinh (2) cua h? tac6:
x 2+ 2 x ( y - l ) + y 2 - 6 y + l = 0, A = ( y - l ) ^ - y ^ + 6 y- 1 = 4y >0 o y >0 Ta viet lai:
y2 + 2 y ( x - 3 ) + x^ - 2 x + l = 0,Á = x 2 - 6 x + 9 - x 2+ 2 x - l = 8-4x>0<=>x<2 Phuong trinh (1) duQ'c viet lai nhu sau: Vx + 1 + ^ x - 1 = ^y"* +2 +
Xet ham so f(t) = t + Vt* +2 tren [0;+oo) ta c6 f'(t)=:l+ >0 nen
71^+2
ham só f(t) dong bien. Lai c6 f ( x - l ) = f(y*) o x = y^ +1. Thay vao phuong trinh thu hai cua h^ ta thu du^-c:
Tdi lifU on thi aai ho, s.hig tao vu xidi fl, bUtl'l, H( fl, mi Ul- K^uycn Tniflg^^
y | y ' ' + 2 y * + y - 4 ) = 0> y = 0
y^ + Zy* + y - 4 = 0
Xet g(y) = y^ + 2y* + y - 4 ta c6 g '(y) = 7y^ + 8ý^ +1 > 0 vai mpi y > 0 nen
ham so g(y) dong bien, ma g(l) = 0 => y = 1 la ngifm duy nhat.
Tir do tim dugc 2 c^p nghi^m ciia h?la: (x;y) = (l;0),(2;l).
b) Ta viet lai phuong trinh thii 2 ciia hf th^nh: A , > 0 Ay 7 l < y < - ^ 3 4 0 ^ x < - 3 x ^ + x ( y - 3 ) + y ^ - 4 y + 4 = 0 y^ + y(x - 4) + x^ - 3x + 4 = 0
The xy = -x^ - y^ + 3x + 4y - 4 tu phuong trinh (2) vao phuong trinh (1) ta thu du(?c: 3x^ + 18x^ + 45x - 3y^ + 3y^ + 8y -108 = 0
+ Xet ham so f(x) = 3x^ + ISx^ + 45x tren -I
nen ham so f(x) dong bien. Suy ra f(x) < f f-1
ta CO f'(x) = 9x^ +6x + 45>0
892 9 + Xet ham so g(y) = -3y^ + 3y^ + 8y -108 tren
g'(y) = 0<=>y = — , t u d 6 d e dang suy ra g(y) < g
3 v3y
taco g'(y) = -9y +6y+8. 892
+ Suy ra f(x) + g(y) < 0. Dau bang xay ra khi va chi khi x = y = — Thir l^i ta tháy cap nghỉm (x;y) = (4_ 4)
3'3 thoa man hf. PHLTONG P H A P D A N H G I A
De giai dup'C h^ phuong trinh bang phuong phap danh gia ta can nim chac cac bat dSng thuc co ban nhu: Cauchy, Bunhicopxki, cac phep bien doi trung gian giiia cac bat ding thuc, qua do de danh gia tim ra quan h$ x,y Ngoai ra ta cung c6 the diing ham so de tim GTLN,GTNN tu do c6 huong danh gia, so sanh phii hgp.
TJty TNHH MTVDVVH KhangVm V i d y 1: G i a i c a c p h u o n g t r i n h s a u 2 a) 1 1 r +
V l + 2x2 ^i + 2y^ y/l + lxy
7 x ( l - 2 x ) + 7 y ( l - 2 y ) = § b) x ( x ^ - y^ ) . x 2 = 2^ ( x- y 2 f 76x2 -20y2 +2 = ^4x{8x + l) Giki a) Dieu ki^n: 0 < x , y < —. Dat a = 72x,b = 7 2 y; a, b 6 0; 4- X Taco: VT = ^<. 2 1 1 - + U + â ' l + b^ J
Ta su dung bo de voi a, b > 0 va ab < 1 ta c6 bat dang thuc:
1 1 2 • + < o ( a - b f ( a b - l ) . / <0 (diing). l + â l + b^ 1 + ab ( l + ab)(l + a 2 ) ( l + b 2 ) Vay VT< r = VF. 7l + ab " ' 0?
DSng thuc xay ra khi x = y . Thay vao(2) ta tim dugc nghỉm ciia phuong trinh. "oil ' 9 - 7 7 3 9-773^ 36 36 9 + V73 9 + ^ ^ ' 36 Nghifm ctia h^ (x;y) =
b) Dieu ki^n: x > y^ > 0.
Phuong trinh (1) tuong duong: x^ + x(x - y^ J - 2 ^ ( x - y ^ = Q. Dat V x- y 2 = u phuong trinh (1) thanh:
x-' + xu^ - 2u^ = 0 X = u <=> y^ = X - x^ .
I Thay vao (2) ta dugc: 96x^ - 20x + 2 = yjsix^ + 4 x .
JTa CO 96x2 _ 7nv ^ ? ^ ^ 9 v 2 ^ 4 v - 3/1 1 ^ 7 0 . 2 I ^ ^^x^ + 4x + 2
0 3( 9 6 x 2-20x + 2)<32x2+ 4 x + 2« ( l 6 x- 2 f < 0 o x = i z * y = ± : ^ .
Tu do ta CO cac nghi^m ciia la: Vay h§ c6 nghi^m (x;y) =
8' 8 V i dv 2: Gidi cac h# phuong trinh sau
2x + V4x^ +1 ](yly^ +1 - y ] = 1 ' a) 1 1 1 • + + - voi x , y > 0 . b) 1 + 3" 1 + 2^ 1 + 5" 1 + 4" Sx + l O ^ x y - y = 12 x + 6 ( x ^ ^ y ^ ) + xy + y^ <3 Giai
i) Ta thay ^y^ +1 - y > y - y S O va f-^/y^ +1 - y A/ y ^ +1 + y = 1 nen phuong trinh (1) ciia h? c6 the viet lai nhu sau: 2x + V 4 x 2+ l = y + 7y^+l phuong trinh (1) ciia h? c6 the viet lai nhu sau: 2x + V 4 x 2+ l = y + 7y^+l Xet ham so f(t) = t +Vt^+ l , f ' ( t ) = l + , * = ^ * J ! I1^ > 0 nen ham so
V t^ + l V t^ + l f(t) dongbieh.
Mgt khac ta c6 f(y) = f(2x) «> y = 2x . Thay vao phuong trinh (2) cua h^ ta c6: ' ' ~ ' ' • 2 1 1 1 + + 3 1 1 •o + • 1 + 3" 1 + 4" 1 + 5" 1 + 4" 1 + 3" 1 + 5" 1 + 4" Do x,y>0=:>3",5" >1. 1 1 2 , ^ » ^, i
Ta CO bo de sau: — + — > voi mpi a,b ^ 1. Th|t v^y bat dang a ^ + l b ^ + l 1 + ab
( a - b f ( a b - l )
thiic can chung minh tuong duong vai: -( l + ab)(l + â — , — ^ 0 . Nhung l + b^) dieu nay la hien nhien dung vai a,b ^ 1. Dau bang xay ra khi va chi khi
a = b = l
1 1 2
Ap dyng vao phuong trinh (*) ta c6: + > 7 = . M|t khac ta
1 + 3 " 1 + 5" i + Vis"
luon CO ^/l5^<^/l6^ = 4" = > — ^ + - ^ > ? = > — ? — . Dau bang
1 + 3 " 1 + 5" 1 + V15" 1 + 4 "
xay ra khi va chi khi x = 0 => y = 0
Tom lai: c6 mpt cap nghi^m duy nhat (x; y) = (O; O)
Cty TNHHMTVDVVHKhang Vijt
b) Theo bat dSng thuc A M - GM ta c6 :
> / ^ ^ ^ ^ 1 2 = 3x + 10V;^-y^3x + 5x + 5 y - y = 8x + 4y=>2x + y > 3 Ta se chung mmh:
i^'^ý) I I 2 2\(x3+y3) ,
x + - f ^ - J 2 x 2 + y2 >2x + y« - J ^ ^ > /2 x 2 + y 2 U x + y ( * ) .
X + xy + y ' ^ ' x + xy + y ' ^ ' Ta c6: x + y < ^2(x^ + y^) De chung minh (•) ta se chung minh bat dang
el - '
thuc manh han la: x 3 + y 3
2 7
x +xy + y^ ^ 2^^(777) (1)
Mat khac ta cung c6: xy < nen (1) se duc?c chung minh neu ta chi
ra dug-c:
. 7^!v^ ^2V2(x^^y^)0 2(x^+y3)>(x2^y2)^/^^777^ ,
x^ +ý^ + i _
2
x^ + y^ + 4x3y3 -3x^y^(x^ + y^) > 0 (2)
Vi y > 0 chia hai ve cho dat t = > 0 bat dMng thiic (2) tro thanh. t^ - 3 t' ' + 4 t 3- 3 t 2+ l > 0
Nhung bát dang thuc nay hien nhien dung do: ' ^ t^ - 3t'^ + 4t^ - 3t2 +1 = (t - l)2(t'' + 2t^ + 2t +1)
• X + -
X +xy + y^ V V >3
Ket hgp tat ca cac van de vua chi ra ta thay chi c6 bp so x, y thoa man dieu
X, y >0 ; , 1
kỉn J2x + y = 3 o x = y = l l a nghi^m ciia h? • ^ , Vi dy 3: Giai cac h? phuong trinh sau
9 i l x 2 + 1 = 3 + 40x
.v.. - vai x,y >0 a) < A T
I 2x + y^
= 3 + 40x
.v.. - vai x,y >0 i"' - • *
b)
x/Zxy + Sx + S = 4 x y - 5 x - 3
Giai