Related Functions

Suppose the graphs of $f(x)$ and $g(x)$ meet the line $x = m$ at points $A, B$ respectively, and meet the line $x = n$ at points $C, D$ respectively, where $m < n$. If the slopes of lines $AC$ and $BD$ are negatives of each other, we call $f(x), g(x)$ $(m, n)$-related.

1. If $f(x)$ and $g(x)$ are both increasing on their domains, prove that no real numbers $m, n$ make $f(x), g(x)$ an $(m, n)$-related pair. 2. Let $f(x) = \mathrm{e}^{ax}$ and $g(x) = ax^2$. If there exist real numbers $m, n$ with $mn > 0$ such that $f(x), g(x)$ are $(m, n)$-related and $|AB| = |CD|$, find the range of possible values of the real number $a$.