Solution
= Solution
For $x>0$, <logarithmic differentiation> gives
$$
\ln f(x)=\frac{\ln x}{x},\qquad
\frac{f'(x)}{f(x)}=\frac{1-\ln x}{x^2}.
$$
Since $f(x)>0$, the <derivative> vanishes precisely when $\ln x=1$. Hence
$$
\boxed{x_0=e,\qquad f(x_0)=e^{1/e}.}
$$
The <derivative> is positive before $e$ and negative after $e$, so this is a <global maximum> on the given positive domain.