Two Crossings and a Product

Let $a > 0$ with $a \neq 1$, and let $f(x) = x^a - a^x$ for $x > 0$.

1. When $a = 2$, find the intervals on which the derivative $f'(x)$ is monotonic. 2. Suppose $y = f(x)$ has exactly two distinct zeros $m > n > 0$.

1. Find the range of possible values of $a$.
2. Prove that $mn > \mathrm{e}^2$.