Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Continuous mathematics / 4 / b / ii

Writing $g=e^{x^{\mathsf{T}}x}$, the gradient and Hessian matrix areConsequently, for any nonzero $v$,In particular the Hessian matrix is a positive semidefinite matrix everywhere, so the Hessian criterion for convexity proves $g$ is convex. The strict inequality along every nonconstant line also proves strict convexity.