Which is larger, $e^\pi$ or $\pi^e$? I don't know how to approach this. 
I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was greater or  $\pi$. And I guess $\pi$ is greater but I don't have a rigorous argument.

Hint: Consider
$$f(x) = x^\frac{1}{x}$$ You can check that $f$ attains a maximum at $ x = e$ and so $$e^{1/e}\geq \pi^{1/\pi}$$
Can you continue from here?
