Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 4 / d / Solution

Positive definiteness makes the quadratic term uniquely maximal atFor each $j$, maximize the concave function $r_j\log {\pi }_j+q_j\log (1-{\pi }_j)$. Its interior critical point isThe same formula gives the appropriate boundary value when $r_j=0$ or $q_j=0$.