Trapping a Critical Point

Let $f(x) = a^x + \mathrm{e}^{-x}$ with $a > 1$.

1. Prove that $f(x)$ has an extreme value. 2. Suppose $f(x)$ attains its extreme value at $x = x_0$. If there exist integers $m, n$ such that $x_0 \in (m, n)$ for every integer $a > 1$, find the minimum value of $n - m$.