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}.\,

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 negative for x < e and positive for x > e, which implies that g(x) (and therefore f(x)) increases for x < e and decreases for x > e. In conclusion, x = e is the unique global maximum of f(x).

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