 Solution

ID: 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/solution

g = e^{x^{T} x}

\nabla g (x) = 2 gx, H (g) (x) = g (2 I + 4 x x^{T}) .

Consequently, for any nonzero

v

v^{T} H (g) (x) v = g (2∥ v ∥^{2} + 4 (x^{T} v)^{2}) > 0.

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.

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