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]

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[edit]

  1. ^ Eric W. Weisstein. "Steiner's Problem". MathWorld. Retrieved 12/08/2010.  Check date values in: |accessdate= (help)