I found a collection of "challenging" calculus questions, and the first is this:
Prove that $$e^\pi > \pi^e$$ Hint: Take the natural log of both sides and try to define a suitable function that has the essential properties that yield [the] inequality.
Taking $ln$ of both sides gets $$\pi > e \cdot \ln(\pi)$$
I thought of replacing the $\pi$ with $\arccos(-1)$ for $$-1 > \cos(e \cdot \ln(\pi))$$ and then maybe $f(x) = \cos(e \cdot \ln(x))$, but I have no idea where to go from there, or if that's the right direction. Can anyone offer hints as to the next steps?
