. x^, 4x +3 y, Ta thay phuong trinh c6 nghỉm khi va chi khi — va
1)Dieu kifn can:
Gia su h$ c6 nghifm (x;y) suy ra (y;x) cung la nghifm ciia h?. Do do de h^ phuong trinh c6 nghiem duy nhát thi y = x
Thay vao h§ ta dugc: Xg - 2XQ + m = 0 pt nay c6 nghiem duy nhát <=> Á = l - m = 0 o m =1
Dieu ki^n dii: Voi m = 1 h? tro thanh: .2 - y + 1 X x = y^ - y + 1 2 + y 2- 2 x - 2 y + 2 = 0 < » ( x - l ) ^ + ( y - l f = 0 < » x = y = l y = x - x + 1
T h u l^i thay thoa man h?. W^y m = 1 la gia trj can tim.
2) Dieu ki^n can: Gia su he c6 nghi#m (xo;yo) suy ra (yo;xo) cung la nghi^, ciia h$. ciia h$.
Do do de h# phuong trinh c6 nghiem duy nhát thi yg = x^ * ' ) ' ' " m
X o = 0
Thay vao he ta dugc: x^ - 5x^ + rnxg = 0(l) -
L x ^ - 5 x o + m = on
(1) CO nghiem duy nhat thi (*) phai v6 nghiem hoac c6 nghiem kep x = 0 ^A = 2 5 - 4 m < 0
<=> A = 2 5 - 4 m = 0 o m > — 4 5 = 0
Dieu kifn du: Voi m > — ta c6:
3 x 2= y( y 2_ 2 y + m) = y [y-lf+m-1
3 y 2= x( x 2- 2 x + mJ = x ( x - l f + m - l
Cpng hai phuong trinh ciia h$ voi nhau ta dugc:
x(x2-5x + m ) + y( y 2 - 5 y + m j = 0 x , y > 0 <=> X X — 2 + m - - 25 5 ' - 2 + m - - 25 = 0 < » x = y = 0 25
Vay m > — la nhiing gia tri can tim.
V i dv 5: Chung minh r5ng h? sau c6 nghifm duy nhát voi mpi a ^ 0: o 2 a
2x = y + — 2 y 2 = x + ^
f Giai DK: X, y ^ 0 . T u hai pt ciia h? suy ra x, y > 0. tuong duong: DK: X, y ^ 0 . T u hai pt ciia h? suy ra x, y > 0. tuong duong:
.2,. ,.2 , ,2 ^ 2 x y ( x - y ) = y 2 - x 2 o ( x - y ) ( 2 x y + x + y) = 0
2y^x = x^ +á
<^x = y (do x,y >0=>2xy + x + y >0).
/i/ / \ / / / ; \;/ V nVVHKhangVilt
Xet ham so f (x) = 2x^ - voi x > 0 .
Taco f ( x ) = 2 x ( 3 x - l ) = ^ f ( x ) = 0 « x = i
Ma f(0) = 0;f ^ 1 ^
27 va â > 0 nen pt(*) c6 d u y nhát mpt nghiem. Vay he da cho luon c6 nghiem duy nhat voi mgi a ^ 0 .
V i d y 6: T i m m de h? phuong trinh sau c6 nghiem duy nhat 2x^y + 2x^ - y ^ - 2 y - m 2xy^ + 4xy - x^ + 2x = m - 1 He p h u o n g trinh Giai 2x^(y + l ) - ( y ^ + 2 y + l ) = m - l 2x(y^ + 2 y + l ) - x ^ + 1 = m 2x2(y + l ) - ( y + l ) ^ = m - l 2x(y + l ) ^ - x^ = m - l D|t y + 1 = t he p h u o n g trinh tro thanh: 2 x ^ . t - t ^ = m - l
2x.t^ - x^ = m - 1
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Dieu kien can: Goi (^XQAQ) la mpt cap nghiem ciia h^
\2x,\t,-t,' = m-\)
[ 2 x o V - V = " ^ - l
He phuong trinh tren la mpt he doi xung nen neu h? c6 nghiem (xo;to) thi h? cung C O nghiem (to;Xo) h? c6 nghiem duy nhat <=> XQ = to thay vao mpt phuong trinh ciia h? ta c6 2XQ - XQ +1 - m (3)
Mat khac vi he c6 nghiem duy nhat nen (3) c6 nghiem duy nhat. Xet f(xo) = 2xo^-Xô + 1 taco
f'(XQ) = 6x^,2 - 2xo = 2xo(3xo - 1 ) = 0 « X o = 0
3 v < f ( x o ) < f ( 0 ) « ^ < f ( x o ) < l
So nghiem cuă3) la so giao diem ciia d o thi ham so y = f(xo) va duong m > 1
thang y = m => (3) CO nghiem duy nhat o m < 26 27
Tom lai dieu ki#n can de h§ c6 nghiem d u y nhat la m > 1 26 m < 27 Dieu kỉn d i i :
Voi m < ^ hoac m > 1 . ta CO h# phuong trinh tuong d u o n g v o i •" ' ' i 27 j(x-t)(2xt + x + t) = 0 rt = x 2 x ^ - t 2 + l = m 2t^ - 1 ^ + 1 = m , r A , (I) • hoac t = - - x 2x + l x^(4x^+2x + l ) = l - m (11) (2x + l ) ^
Voi m > 1 t h i (I) c6 nghiem d u y nhat va (II) v6 nghiem nen h? c6 nghiem duy nhat.
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Voi < ^ thi (I) C O mpt nghiem d u y nhat. Lap bang bien thien ta suy ra nghiem do la nghiem am.
x^(4x^ + 2x +1)
Xet ham so f(x) = — — - 1 + m . La ham lien tuc tren [0: +oo)
(2x + l)2
Mat khac ta c6 f(0) = m - l < 0 ; l i m f(x)=::+oo nen p h u o n g trinh f(x) = 0 c6
X-»+QO
it nhat mpt nghiem d u o n g . Do do h? (II) c6 it nhat mot nghiem. V i vay h? ban dau c6 it nhat 2 nghiem.
26 T- !
Do do m < — khong thoa man yeu caụ j
Tom lai de h? c6 nghiem d u y nhat thi m G (1; +oo) 2) Dieu kif n de c6 k nghif m
Khi gap cac bai toan dang nay ta thuong lam theo each: Dat dieu k i ^ n ciia 1 an (neu c6)
' j Bieu dien mpt an theo an con lai
Dua he p h u o n g trinh ve dang 1 phuong trinh tuong d u o n g sau d o tim dieu
ki^n de p h u o n g trinh c6 k nghiem xet cac vi dy sau:
du 1: Tim m de he phuong trinh: J>han bi^t.
3(x + l ) ^ + y - m = 0
^ C O 3 cap nghỉm X + (adsbygoogle = window.adsbygoogle || []).push({});
G i a i x < l y = - - 2x + 1 (do X = 0 k h o n g la nghi^m Ta c6: x + yjxy = 1 <=> .^xy = 1 - x <=> p h u a n g trinh) — 2x + 1
Thay vao p h u o n g t r i n h t h i i nhat ta dup-c: 3x + 6x + = m - 3 (a) x