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

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

so $g^{\prime }(x)$ 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", Crelle 40: 208, https://www.digizeitschriften.de/download/PPN243919689_0040/PPN243919689_0040___log28.pdf