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e is the global maximum of x root x, 2 root 2 = 4 root 4, so...
#3
Oh, now with the picture I see what you mean.

To get a formula we can use the Lambert W function.
The Lambert W function is the inverse of the function M(x)=x*e^x
and we can express the self root x^(1/x) with help of M:

y = x^(1/x) = exp(-M(-ln(x)))

You can verify this with a bit of calculation.
Then we can obtain the inverse:

exp(-W(-ln(y))) = x

More exactly x can be two values, left and right from e, which correspond to the two branches of W:




So when you want to get the left value - as in your case - you choose x_L and get:

would be simply 3 again.
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Messages In This Thread
RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by bo198214 - 02/13/2011, 10:27 AM

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