Steiner's calculus problem

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

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

It is named after Jakob Steiner.

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

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

Applying the first derivative test, the derivative of $g$ is

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

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

References

• Steiner, J. (1850), "Über das größte Product der Theile oder Summanden jeder Zahl" (PDF), Journal für die reine und angewandte Mathematik, 40: 208