$f : [0,1] \to \mathbb R$ is continuous, and $f(0) = f(1) = 0$. $f''$ exists and $f''(x) ≥ 0$ at all $x \in (0,1)$. Show that $f(x) \le 0$ for all $x \in (0,1)$.
- Use Rolle's Theorem, there exists point $c$ such that $f'(c)=0$.
- Since $f''(x) \ge 0$, then $f'(x)$ strictly increasing.
- By 1 + 2, $f$ attains minimum at $c$. (How to show that ?)
- Stuck here, though I can visualize the graph in my mind. By contradiction?