Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 30K / a / Solution

If $y_1\le y_2$, then the utility function is increasing, so for every $X\in \mathcal{X}$,Taking suprema gives $F(y_1)\le F(y_2)$.

For $0\le t\le 1$, let $X_1$ and $X_2$ attain the suprema at $y_1$ and $y_2$. Because $\mathcal{X}$ is a vector space,The concavity of $U$ now givesThus $F$ is increasing and concave, as stated by optimized affine shift of concave utility.