SE-SKOLORNAS 1965 Problem 5

Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $-1 \le x \le 1$ . Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.