Elementary inequality: $x^x+y^y \geq x^y +y^x$?

Question

Let $x,y>0$, it seems (with numerical simulations) that $x^x+y^y \geq x^y +y^x$.

If this is true, it has to be well known by some people.

Does this inequality have a name? several proofs?

Answer 1

Fix $y > 0$, and let $$f(x) = x^x - x^y - y^x.$$ We are done if we can show the minimum of the $f$ is $-y^y$. Computing the derivative, $$f'(x) = -y x^{y-1}+x^x (\log (x)+1)-y^x \log (y).$$ $f'$ has a zero when $x = y$ and changes from positive to negative there. Thus $f$ has a local minimum. But $f$ increases for $x > y$, so $f(y) = -y^y$ is the global minimum.