Steiner's problem

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Steiner's problem is the problem of finding the maximum of the function

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

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 12/08/2010.

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