Show that $e^{\pi}>\pi^e$.

Question

Show that $e^{\pi}>\pi^e$.

Note that you don't know $e=2.7182...$ nor $3.1415...$. How can we show this?

Thank you.

p.s. This is just a question came upin my mind sudden;y, it may not have a nice solution.

Answer 1

Hint: Consider the function $f(x)=x^{1/x}$. It has a maximum value.