Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 16 / e / Solution

The interior stationary point of the probability density function solves $12x(2-3x)=0$ with $0<x<1$, giving $x=2/3$. The sign change of $f^{\prime }$ established in part (d), together with zero endpoint values, makes this the global maximum. Thus