Suppose are any positive real numbers such that Prove that:

Source: JIPAM

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- Mar 31st 2009, 02:23 AMNonCommAlgPre-algebra: inequality
Suppose are any positive real numbers such that Prove that:

__Source__: JIPAM - Mar 31st 2009, 01:55 PMbkarpuz
We are looking for where under the restriction .

Hence, we set , therefore it suffices to find .

Considering the symmetricity, and it suffices to examine the partial derivative of with respect to .

It follows that the critical points are and .

Clearly, .

On the other hand, let

We find that the critical point for is , and , which is the max value of (at ) at the same time because the other critical point makes attain .

Therefore, the given inequality holds.

**Not**. I know this is a long solution and not so nice. :S - Mar 31st 2009, 04:04 PMNonCommAlg
- Mar 31st 2009, 09:38 PMbkarpuz
(Doh) I know, as I know in JIPAM there are so mant inequalities using convex functions, I guess u wish to see a solution in that direction. But I have no idea since I am not focused on this subject. (Speechless)

I just wanted to share my long and poor solution. (Giggle)