Question

Prove that: (is true for any positive x,y):? `x^x*y^y>=((x+y)/2)^(x+y)`

`x^x*y^y>=((x+y)/2)^(x+y)`

Answer 1

See below.

Explanation:

Consider `f(x) = x ln x`

This function has a convex hypograph because

`f''(x) = 1/x > 0`

so in this case

`f((x+y)/2) le 1/2(f(x)+f(y))` or

`((x+y)/2)ln((x+y)/2) le 1/2( x ln x + y ln y)` or

`((x+y)/2)^((x+y)/2) le (x^x y^y)^(1/2)`

and finally squaring both sides

`((x+y)/2)^(x+y) le x^x y^y`