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Topic: y = x^(1/x)
Replies: 14   Last Post: Jun 12, 2001 1:42 AM

 Messages: [ Previous | Next ]
 Martin Cohen Posts: 55 Registered: 12/8/04
Re: y = x^(1/x)
Posted: May 29, 2001 4:32 PM

James wrote:
>
> Whilst messing about on my graphical calculator, I found that the graph of
> the xth root of x:
>
> y = x^(1/x)
>
> peaks when x = e.
> Just out of interest, why is this? Can anyone offer proof? I know that
> this is probably baby stuff for most of you... sorry if I waste your time :)
>
> Thanks,
> James

I saw this proof a number of years ago - can't remember for sure where
(maybe in "What is Mathenatics").

The only assumption is that e^x >= 1+x for all real x
with equality only when x = 0.

Substitute (x-e)/e for x. Then

e^((x-e)/e) >= 1 + ((x-e)/e) = x/e

or e^(x/e - 1) >= x/e

or e^(x/e) >= x

or e^(1/e) >= x^(1/x)

with equality only when x = e.

Martin Cohen

Date Subject Author
5/27/01 abc
5/27/01 gyan doshi
5/28/01 Dave L. Renfro
5/28/01 David W. Cantrell
5/29/01 Dave L. Renfro
5/30/01 David W. Cantrell
5/30/01 Dave L. Renfro
5/27/01 Marko Marin
5/27/01 Norm Dresner
5/27/01 John Prussing
5/28/01 Paul V. S. Townsend, M.Sc.
5/28/01 David W. Cantrell
5/28/01 Oscar Lanzi III
5/29/01 Martin Cohen
6/12/01 Bill Taylor