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!

This is what is called a "happy idea". Sometimes the step between the known ($e^x \geq x+1$) and the unknown ($e^\pi > \pi^e$) is not obvious but some techniques exist that can help. I suggest you read "How to solve it" by G.Pólya for some of those techniques. — Darth Geek, Jul 30 '14 at 13:50
Huh, I had always maximized the function $x^{e\pi \over x}$. — zibadawa timmy, Jul 30 '14 at 16:30

score 3 · Accepted Answer · edited Apr 13 '17 at 12:20

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.

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

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