Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 3 / b / Solution

For $x>0$, logarithmic differentiation givesSince $f(x)>0$, the derivative vanishes precisely when $\ln x=1$. HenceThe derivative is positive before $e$ and negative after $e$, so this is a global maximum on the given positive domain.