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 natural logarithms. One can determine that by solving the equivalent problem of maximizing

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

The derivative of ${\displaystyle g}$ can be calculated to be

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

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

References

1. Eric W. Weisstein. "Steiner's Problem". MathWorld. Retrieved December 8, 2010.

• Steiner, J. (1850), "Über das größte Product der Theile oder Summanden jeder Zahl" (PDF), Crelle, 40: 208