Let $f \colon [-1,1] \to \mathbb R$ be a twice differentiable function s.t. $f(-1)=f(1)=0$ and there exists $k>0$ s.t. $\vert f''(x) \vert \le k$ for every $x \in [-1,1]$. Show that $ \max_{[-1,1]}\vert f \vert \le \frac{k}{2}. $
The book suggests: take $x_0\in (-1,1)$ such that $\vert f(x_0)\vert$ is maximum and use Taylor.
Following this hint, I write: $ f(x_0+h)=f(x_0) + \frac{f''(\xi)}{2}h^2 $ The linear term, $f'(x_0)=0$ (since the point is in the interior of $[-1,1]$ and is a maximum or a minimum for $f$).
Now how can I conclude? I can't see how to use the hypotesis $f(-1)=f(1)=0$.