How to prove $\pi^3<3^\pi$ without using explicit value of $\pi$?

Question

How to prove $\pi^3<3^\pi$ without using explicit value of $\pi$? In the following link I proposed similar problem and got extremely amazing explanations which uses pure geometrical ideas : How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$

Here too I am thinking some similar kind of idea (not involving calculus) but not getting any. Can there also be something like that?

$f(x)=x^{1/x}$ is increasing when $x\in(0;e]$, decreasing when $x\in [e;+\infty)$. Since $e<3<\pi$, we get the result. — user263326, Feb 08 '17 at 14:58

score 4 · Answer 1 · edited Apr 13 '17 at 12:20

4

Take the logarithm. Divide each side by $3 \pi$. Notice the question becomes:

$$ \frac{\ln \pi}{\pi} < \frac{ \ln 3}{3}$$

However, the derivative of $ f(x)=\frac{\ln x}{x}$ is $$f'(x)=\frac{1-\ln x}{x^2}$$ From the product rule. So the fucntion $f(x)=\frac{\ln x}{x}$ is decreasing if $x>e$ as $f'(x)<0$. The result follows as. $$\pi >3>e$$As seen here, here and here.

edited Apr 13 '17 at 12:20

Community

1

answered Feb 08 '17 at 14:44

S.C.B.

22,768

And well, that is a lot of links! – S.C.B. Feb 08 '17 at 14:47
I want to use method that uses geometry like straight line is the shortest path between any two points. – Subhash Chand Bhoria Feb 08 '17 at 15:21
@SubhashChandBhoria You're going to have some very bad time trying to prove it using only geometric methods - how are you imagining a geometric interpretation of $3^\pi$? – Wojowu Feb 08 '17 at 15:26
@SubhashChandBhoria I'm with Wojowu on this one. You can't geometrically interpret something like $3^\pi$. – S.C.B. Feb 08 '17 at 15:36
that's what i'm trying – Subhash Chand Bhoria Feb 10 '17 at 03:52
@SubhashChandBhoria Well good luck then. – S.C.B. Feb 10 '17 at 04:00

How to prove $\pi^3<3^\pi$ without using explicit value of $\pi$?

1 Answers1