Steiner's calculus problem

Steiner's problem, asked and answered by Steiner (1850), is the problem of finding the maximum of the function

${\displaystyle f(x)=x^{1/x}.\,}$^[1]

It is named after Jakob Steiner.

The maximum is at ${\displaystyle x=e}$, where e denotes the base of the natural logarithm. One can determine that by solving the equivalent problem of maximizing

${\displaystyle g(x)=\ln f(x)={\frac {\ln x}{x}}.}$

Applying the first derivative test, the derivative of ${\displaystyle g}$ is

${\displaystyle g'(x)={\frac {1-\ln x}{x^{2}}},}$

so ${\displaystyle g'(x)}$ is positive for ${\displaystyle 0<x<e}$ and negative for ${\displaystyle x>e}$, which implies that ${\displaystyle g(x)}$ – and therefore ${\displaystyle f(x)}$ – is increasing for ${\displaystyle 0<x<e}$ and decreasing for ${\displaystyle x>e.}$ Thus, ${\displaystyle x=e}$ is the unique global maximum of ${\displaystyle f(x).}$