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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]

Mplwp Steiners problem

It is named after Jakob Steiner.

The maximum is at x 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 {\displaystyle g} 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 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" (PDF), Crelle, 40: 208

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