IMC 1998 · Problem 2

Let

$$\mathcal{P} = \left\{ f \;:\; f(x) = \sum_{k=0}^{3} a_k x^k, \; a_k \in \mathbb{R}, \; |f(\pm 1)| \le 1, \; \left| f\left(\pm \tfrac{1}{2}\right) \right| \le 1 \right\}.$$

Evaluate

$$\sup_{f \in \mathcal{P}} \; \max_{-1 \le x \le 1} |f''(x)|$$

and find all polynomials $f \in \mathcal{P}$ for which the above "sup" is attained.