(a) Let f(x) = x^(1/x) for all x>/=1. Find the greatest value of f(x).
(b) By (a), find a possitive interger m, such that m^(1/m) >/= n^(1/n) for all positive intergers n.
y = x^(1/x)
y' = -1/x^2 x^(1/x)(lnx-1)
Since x>=1, y'=0 when lnx = 1, or x=e.
y(e) = e^(1/e) =~ 1.44
Since f is decreasing for x > e e^(1/e) >= n^(1/n) for n > e. Since we want integers, we need to check f(x) for x = 1,2,3 to find which has the greatest f(x). That will be m.