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 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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