Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 7H / c / Solution

Let $z=tx+(1-t)y$. Apply the hypothesis with first argument $z$ and second argument successively $x,y$:Multiply these inequalities by $t$ and $1-t$ respectively and add. The right sides cancel, leaving $f(z)\le tf(x)+(1-t)f(y)$.