(a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex and strictly convex.

(b) State a necessary and sufficient first-order condition for strict convexity of $f \in C^{1}\left(\mathbb{R}^{n}\right)$, and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.