黄慕兰子女现状:已知两正数x，y满足x+y=1，求证：(x+1/x)^2+(y+1/y)^2>=25/2

来源：百度文库编辑：查人人中国名人网时间：2024/05/03 08:08:58

as x + y = 1, we have 2*sqrt(xy) <= x + y = 1
thus xy <= 1/4 (when x = y = 1/2, '=' is valid)
so x^2 + y^2 = 1 - 2xy >= 1/2 ----------- (1)
and 1/(xy)^2 >= 1/(1/4)^2 = 16 ----------- (2)

then we have original expression:
x^2 + 2 + 1/x^2 + y^2 + 2 + 1/y^2
= x^2 + y^2 + (x^2 + y^2)/(xy)^2 + 4
as (1) and (2), the expression above can be rewritten as
......
>= 1/2 + 16/2 + 4 = 25/2

so the required inequality is achieved and equality is valid when and only when x = y = 1/2

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