Past exam of the optimisation for data science course of the University of Oxford / 2023 / 2 / a

Let $d=y-x$ and take $0<t\le 1$. Rearrange the defining inequality for a strongly convex function, applied to $x+td$:Because $f$ is differentiable at $x$, its directional derivative in direction $d$ isTaking the limit in the rearranged inequality givesThis is the supporting affine function inequality strengthened by a positive squared Euclidean distance. Differentiability at $x$ alone suffices; no derivative at $y$ is used.