Steiner's Problem

For what value of $x$ is $f(x)=x^{1/x}$ a Maximum? The maximum occurs at $x=e$, where

$$f'(x)=x^{-2+1/x}(1-\ln x)=0,$$

which gives a maximum of

$$e^{1/e}=1.444667861\ldots.$$

The function has an inflection point at $x=0.581933\ldots$, where

$$f''(x)=x^{-4+1/x}[1-3x+(\ln x)(2x-2+\ln x)]=0.$$

See also Fermat's Problem