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}}.\,

^[1]

It is named after Jakob Steiner.

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

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

The derivative of $g$ can be calculated to be

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

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

References

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

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

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