Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 29K / c / Solution

Differentiating $f(\mu ,\sigma )=\mathbb{E}[U(\mu +\sigma Z)]$ twice gives the Hessian matrixFor every vector $(a,b)$,Thus the Hessian is a negative semidefinite matrix, andEquivalently, with $W=-U^{\prime \prime }(X)\ge 0$, its determinant isby the Cauchy-Schwarz inequality, while both diagonal entries are nonpositive.