TITLE: Find the maximum of $f(x)=x^{1/x}$
QUESTION [7 upvotes]: Find the maximum of the function $$f(x)=x^{1/x}$$ and the value of $x$ which gives the maximum value?

REPLY [24 votes]: This is not original with me.
If we know that $e^x \ge 1+x$
with equality only when $x = 0$,
$e^{(x-e)/e} \ge 1 + (x-e)/e = x/e$
or
$e^{x/e} \ge x$
or
$e^{1/e} \ge x^{1/x}$
with equality only if $x = e$.
Ta-dah!
At no time do the fingers leave the hands!