Comparing $\pi^e$ and $e^\pi$

Question

Comparing $\pi^e$ and $e^\pi$ without calculating them

I read the answer there but I didn't understand one thing. How I should know to put $\dfrac{π}e-1$ instead of $x$? If I had this question on a test, I had no idea what to put instead of $x$. I mean, why the first thing I need to think about is to calculte when $x=\dfrac{π}e-11$ .

I hope you understand my question.

Note :

This is not a duplicate - I'm not asking what is bigger - I don't understand the answer, that's all!

Answer 1

Note: Your question is less about whether you understand the proof, and more about how you would construct such a proof.

In a test, you wouldn't be expected to find that argument, which is a slick simple argument that was found by somebody probably with a little extra time, trying to find a proof without using general properties about derivatives.

The usual argument is the answer given by Yuval Filmus.