Why does $\tan\left(x ^ {1/x}\right)$ have a maximum value at $x=e$?

Question

Why does this function, $$\tan\left(x ^ {1/x}\right)$$ have a maximum value at $x=e$?

Answer 1

The maxima of a continuous function can be found when the derivative is equal to zero. In this case, $$ f(x) = \tan{(x^{\frac{1}{x}})} \qquad \Rightarrow \qquad f'(x) = \left(-x^{1/x - 2}\right) \left( \ln{x}-1\right)\left(\sec^2{\sqrt[x]{x}} \right) $$ (you can ask Wolfram Alpha for this.) For this to be zero, the only option is $\ln{x}-1 = 0 \Rightarrow x= e^1 = e$.

Answer 2

Here $\tan$ is a red herring. For any strictly monotonous $C^1$-function $g$ it holds that $x^{1/x}$ and $g\bigl(x^{1/x}\bigr)$ share their extrema at the same values.

To get the extrema in our case replace $\tan$ by $\ln$ and happily find them.