## (a^a)(b^b) + (a^b)(b^a) ≤ 1

Let   $a$   and   $b$   be positive real numbers such that   $a \; + \; b \; = \; 1$.

Prove that

$a^a \, b^b \; + \; a^b \, b^a \; \leq \; 1$