Steiner's Calculus Problem

Theorem
Let $f: \R_{>0} \to \R$ be the real function defined as:
 * $\forall x \in \R_{>0}: \map f x = x^{1/x}$

Then $\map f x$ reaches its maximum at $x = e$ where $e$ is Euler's number.

Proof
$\dfrac {x^{1/x} } {x^2}$ is always greater than $0$.

Therefore:


 * $\map {f'} x > 0$ for $\ln x < 1$
 * $\map {f'} x = 0$ for $\ln x = 1$
 * $\map {f'} x < 0$ for $\ln x > 1$

By Derivative at Maximum or Minimum, maximum is obtained when $\ln x = 1$,

that is, when $x = e$.