Equality without calculator

Question

$ e^\pi$ and $\pi^e $ Which one is greater, How do I proceed without a calculator, I do not have any idea how to solve , using AM GM concept

Answer 1

Consider function $f(x)=\frac{\ln x}{x}$

$f'(x)=\frac{1-\ln x}{x^2}$

$f'(x)$ is negative for $x>e$ that is the function is reducing in this interval.Since $e<\pi$ so we have:

$\frac{\ln e}{e}>\frac{\ln \pi}{\pi}$ ⇒ $e^{\pi}> \pi^e$