Question

若实数x,y满足x>=y>0,且x=4y^(1/2)+2(x-y)^(1/2),则x的取值范围是

若实数x,y满足x>=y>0,且x=4y^(1/2)+2(x-y)^(1/2),则x的取值范围是

Answer 1

实数x,y满足x>=y>0,且x=4y^(1/2)+2(x-y)^(1/2)，
∴x^2=16y+4(x-y)+16√[y(x-y)]
=12y+4x+16√[y(x-y)],记为f(y),
f'(y)=12+8(x-2y)/√(xy-y^2)=0,
3√(xy-y^2)=2(2y-x),
平方得9(xy-y^2)=4(4y^2-4xy+x^2),
整理得25y^2-25xy+4x^2=0,
y1=4x/5,y2=x/5,
f(x)=16x,f(4x/5)=20x,f(x/5)=64x/5,f(0)=4x,
∴4x<x^2<=20x,x>0,
∴4<x<=20,为所求.

Answer 2

实数x,y满足x>=y>0,且x=4y^(1/2)+2(x-y)^(1/2)，
∴x^2=16y+4(x-y)+16√[y(x-y)]
=12y+4x+16√[y(x-y)],记为f(y),
f'(y)=12+8(x-2y)/√(xy-y^2)=0,
3√(xy-y^2)=2(2y-x),
平方得9(xy-y^2)=4(4y^2-4xy+x^2),
整理得25y^2-25xy+4x^2=0,
y1=4x/5,y2=x/5,
f(x)=16x,f(4x/5)=20x,f(x/5)=64x/5,f(0)=4x,
∴4x<x^2<=20x,x>0,
∴4<x<=20,为所求.