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Re: y = x^(1/x)
Posted:
May 29, 2001 4:32 PM
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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
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