Question

已知实数x,y满足x^2+y^2=2x,求x^2y^2取值范围 用导数知识解答。

Answer 1

x^2+y^2=2x
（x-1)^2+y^2=1
令x=sina+1,y=cosa
xy=(sina+1)cosa=sinacosa+cosa
(xy)'=(cosa)^2-(sina)^2-sina
=-2(sina)^2-sina+1
=-(sina+1)(2sina-1)
令(xy)'=0
sina=-1或sina=1/2
-1≤sina≤1/2时,(xy)'≥0,函数单调递增；
sina=-1时,有(xy)min=0 sina=1/2时,有(xy)max=3√3/4
1/配粗2≤sina≤巧卖陵1时,（xy)'≤0,函数单调递减.
sina=1/2时,有(xy)max=3√3/4 sina=1时,有(xy)min=0
因此
0≤x^2y^2≤（3√3/4)^2
0≤x^2y^2≤27/16
x^2y^2的取值范围孝戚为[0,27/16]