$e^ \pi > \pi ^ e$ .

Prove that $e^ \pi > \pi ^ e$ .

This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

#Calculus #Proofs

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Take the function $f:\mathbb{R_+}\mapsto \mathbb{R}$ , defined by : $f(x)=\frac{\log x}{x}$ .

It derivative is : $f'(x)= \frac{1-\log x}{x^2}$ , therefore $f(e)$ is the maximum.

Then : $f(e)>f(\pi)$ , which means that : $\pi \log e >e\log \pi \Leftrightarrow e^{\pi} >\pi^e$ .

How did I choose this function : just try to get $\pi$ is one side and $e$ is another to get the idea.

Haroun Meghaichi - 7 years, 2 months ago

Brilliant! Great solution.

Carlos E. C. do Nascimento - 7 years, 2 months ago

This solution exists in the math book of highschool's terminal year in Algeria. I loved it ;)

Zakaria Sellami - 7 years, 1 month ago

I took the function $f(x) = x^{\frac{1}{x}}$ and found that it has a maximum at $x = e$ .So $f(e) > f(\pi)$ . $e^{\frac{1}{e}} > \pi^{\frac{1}{\pi}}$ . $e^{\pi} > \pi^{e}$ .

Eddie The Head - 7 years, 2 months ago

This is the way I did it a long time ago. This one is an oldie. But this approach takes longer than Haroon's proof.

Michael Mendrin - 7 years, 2 months ago

Yeah..I also solved this one a long time ago ....before i joined Brilliant!!

Eddie The Head - 7 years, 2 months ago

@Eddie The Head – Hey, me too.

Michael Mendrin - 7 years, 2 months ago

A well-known and important inequality: $e^n>1+n$ given $n>0$ . Now if we let $n=\dfrac \pi e-1$ we will immediately get after rearranging our inequality $e^\pi>\pi^e$ .

حكيم الفيلسوف الضائع - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

eπ>πe e^ \pi > \pi ^ e eπ>πe.

Comments

$e^ \pi > \pi ^ e$ .