Let $f: [a,b] \to \mathbb{R}$ with class $C^2$. Prove there exists $M \ge 0$ such that $g(x) = f(x) + M x^2$, where $x \in [a,b]$, is convex.
I get that $g''(x) = f''(x) + 2M$ to show that $g$ is convex I need to show that $g''(x) \ge 0$, as $M \ge 0$ I just need to prove that $f''(x) \ge 0$, but how do I get that?