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Tough question for the good mathematicians watch

1. Find the maximum value of n^(1/n) and prove that it is indeed the maximum

I think basically youve got to find a differentiate it in some form.

Ive got it to xlny=lyx and y^x=x

I dont know if this helps, any ideas anyone
2. (Original post by r_perry)
Find the maximum value of n^(1/n) and prove that it is indeed the maximum

I think basically youve got to find a differentiate it in some form.

Ive got it to xlny=lyx and y^x=x

I dont know if this helps, any ideas anyone
n=e^ln n
=>n^(1/n) = (e^ln n)^1/n =e^((ln n)/n)
derivative of e^f(n) = f'(n)e^f(n)
derivative of (ln n)/n = [1 - (ln n)(-n^-2)]/n^2 = [1+(ln n)/n^2]/n^2
=>derivative of n^(1/n) = [[1+(ln n)/n^2]/n^2]e^((ln n)/n)
then set that equal to 0 and then show that its a maximum by differentiating again or by drawing a graph
3. *has no idea what he's doing in a thread entitled "for the good mathematician" *

Good luck mate!!
4. (Original post by r_perry)
Find the maximum value of n^(1/n) and prove that it is indeed the maximum

I think basically youve got to find a differentiate it in some form.

Ive got it to xlny=lyx and y^x=x

I dont know if this helps, any ideas anyone
Maximum value occurs when n = e

y = n^(1/n) = e^[(1/n)(ln n)]
y' = [(1-ln n)/n^2]e^[(1/n)(ln n)]
y' = 0 => (1-ln n)/n^2 = 0 => (1-ln n) = 0 => ln n = 1 => n = e

To prove its a maximum, differentiate y twice and set n = e and notice it will be < 0:

y'' = n^(1/n)*(-ln(n)/(n^2) + 1/(n^2))^2 + n^(1/n)*(2*ln(n)/(n^3) - 3/(n^3))

y''(e) = [-e^(1/e)]/(e^3) = -0.7193 hence is a maximum QED

Euclid.

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