Creative Calculus Proof that I can't manage

Asked Jun 13 '24 at 17:12

Active Jun 13 '24 at 20:51

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I was working on a Calculus problem set that was supposedly very challenging, and indeed it was, as the first problem I was already stuck on. Here it is:

Prove that $e^{\pi}$ > ${\pi}^{e}$. The hint was to find a suitable function with the essential properties that yield the inequality.

I took the natural log of both sides and got ${\pi}$ * ln(e) = e * ln(${\pi}$). I managed more algebraic manipulation and tried some limits, but nothing has gotten me even remotely close. I don't need an answer; I need a nudge in the right direction.

Also, if my formatting sucks, comment on that, and feel free to edit and correct the post.

edited Jun 13 '24 at 20:51

Pustam Raut

2,490

asked Jun 13 '24 at 17:12

Garrett Zhou

2

Well, you know what $\ln(e)$ is – Deane Jun 13 '24 at 17:14
Yes, that was part of the algebraic manipulation I did. – Garrett Zhou Jun 13 '24 at 17:16
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By the way, the equals sign does not belong there since the two sides are not equal – Deane Jun 13 '24 at 17:17
are you allowed to take derivatives? – Thomas Jun 13 '24 at 17:21
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If you denote $f(x) = x^{\frac1{x}}$ then it is sufficient to compare $f(\pi)$ and $f(e)$ which should be manageable by calculus. – Sahaj Jun 13 '24 at 17:49
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Your problem is equivalent to proving that $\frac{\ln\pi}{\pi} < \frac{\ln e}{e}$, that is why you may study the function $\frac{\ln x}{x}$, which turns out to reach its maximum at $x = e$. – Abezhiko Jun 14 '24 at 07:18

Creative Calculus Proof that I can't manage

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