Prove or disproe:

For any positive integer n and

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- Mar 4th 2010, 07:24 PMelimAn Inequality? (A Challenging One: Prove or disprove)
Prove or disproe:

For any positive integer n and - Mar 4th 2010, 11:49 PMSudharaka
- Mar 5th 2010, 08:39 AMelimSome how I just could not see this clearly before
Say n (>1) is the smallest integer such that for some minimal point ,

,

then

On the other hand the critical points for the function are

They are the points in such that

(note that by our finding above)

But

Thus we get a contradiction and we actually proved that

This is a bit stronger statement.

Thanks Sudharaka for the direction. I thought this problem needs more advanced tech to solve. - Mar 5th 2010, 09:19 AMSudharaka
Dear elim,

We have to show that for any positive integer n and

When n=1,

Therefore the expression is true for n=1

Suppose the expression is true for n=p where

Then,

Now,

Therefore,

Hence,

Therefore our the expression is true for

Hence by mathamatical induction the expression is true for all

Hope this will help you. - Mar 5th 2010, 11:29 AMelim