Which is bigger: or ?

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- Jan 14th 2010, 06:57 PMBraveHeartWhich is bigger: π^e or e^π? Prove it.
Which is bigger: or ?

- Jan 14th 2010, 07:41 PMBacterius
We know that .

Let us conjecture that .

This means that , that is, . Therefore, if our conjecture is true, we must have .

Let us consider the function . Find its minimum on . You will find that the minimum of is , and this minimum is reached when .

But we know that , thus the minimum is never reached with .

. You can follow the steps backwards to finally prove that

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That was the nearly-complete proof. Now, you only have to show that the minimum of is when , and you are done :) - Jan 14th 2010, 07:57 PMDrexel28

Here is another way I just thought of.**Bacterius**just showed us that this is equivalent to showing that ...BUT! this is the same as

(Evilgrin) and then we have to do is show that for but this is clear since for this interval we have that

and so . TA-DA! No calculations needed! - Jan 14th 2010, 07:58 PMBacterius
Lol, I wasn't sure my proof was going to pass since this is the algebra forum and my proof is already half-way into calculus, but Drex actually brought in the integral ! :D

Nice proof though, everything is already trivially proved and one just has to put the bits together :) - Jan 14th 2010, 08:01 PMJhevon
- Jan 14th 2010, 08:04 PMBacteriusQuote:

I'd like to see someone use pre-university algebra to solve this one!

- Jan 14th 2010, 08:19 PMDrexel28
Another way to do it using derivatives is let . We have that and it's easy to show it's a minimum. So then .

- Jan 14th 2010, 09:38 PMChris L T521
See here.

- Jan 15th 2010, 01:09 AMUnbeatable0
Another one:

Note that this can be generalized: for all , it is true that - Jan 15th 2010, 02:45 PMCaptcha
- Jan 15th 2010, 03:22 PMPlato
- Jan 16th 2010, 12:08 AMUnbeatable0
- Jan 16th 2010, 02:15 PMJhevon
graphing actually occured to me, but i don't like this since actually graphing that graph without the use of calculus would be a pain. if you asked a pre-university student to do this, they'd probably just use a graphing utility or plot points with a calculator, in which case they wouldn't need the graph to find the answer to the problem, they'd just use the calculator. i think the idea here is to manually do the problem.

- Jan 17th 2010, 08:00 AMI4talent