Consider the function $f(n) = \Big( \dfrac{1}{n} \Big)^n$.
By setting $f'(n) = 0$, we find that the maximum of $f(n)$ occurs at $n = \dfrac{1}{e}$. Going through the calculations, there doesn't seem to be any "reason" for the appearance of $e$.
However, we do know that the definition of $e$ is given by $$ e:= \lim_{n \to \infty} \Big( 1 + \dfrac{1}{n} \Big)^n. $$
Is there any chance the similar forms of the definition of $e$ and $f(n)$ provide an "explanation" for $n = \dfrac{1}{e}$ maximizing $f(n)$? Is there any way we could just look at $f(n)$ and intuitively "see" that it has to have its maximum at $n = \dfrac{1}{e}$? Or is this just a mathematical "coincidence?"