Steiner's calculus problem

Steiner's problem, asked and answered by, 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 the natural logarithm. One can determine that by solving the equivalent problem of maximizing


 * $$g(x) = \ln f(x) = \frac{\ln x}{x}.$$

Applying the first derivative test, the derivative of $$g$$ is


 * $$g'(x) = \frac{1-\ln x}{x^2},$$

so $$g'(x)$$ is positive for $$0e$$, which implies that $$g(x)$$ – and therefore $$f(x)$$ – is increasing for $$0e.$$ Thus, $$x=e$$ is the unique global maximum of $$f(x).$$