Question

设x,y均大于0,且x+y=1,求证:(x+1/x)∧2+(y+1/y)∧2≥25/2

Answer 1

因为x + y = 1,所以 2根号(xy) = 1/(1/4)^2 = 16 ----------- (2)

原式左边=

x^2 + 2 + 1/x^2 + y^2 + 2 + 1/y^2

= x^2 + y^2 + (x^2 + y^2)/(xy)^2 + 4

根据(1),(2)可得

>= 1/2 + 16/2 + 4 = 25/2

即(x+1/x)^2 + (y+1/y)^2 >= 25/2