Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 4 / a / Solution

Put $\mathbf{h}=\mathbf{y}-\mathbf{x}$. For $0<t\le 1$, convexity gives $f(\mathbf{x}+t\mathbf{h})\le (1-t)f(\mathbf{x})+tf(\mathbf{y})$, henceBy the first-order form of Taylor's theorem, the right side tends to ${\mathbf{h}}^{\mathsf{T}}\nabla f(\mathbf{x})$ as $t\downarrow 0$. ThereforeThis is the supporting inequality for a differentiable convex function. The stated partial derivatives are enough: coordinate derivatives give directional Taylor expansion for convex functions, so no unmentioned continuity assumption on the second partial derivatives is needed. The result in fact needs less smoothness than the question assumes.