ISL 1982 Problem 56

Let $f(x) = ax^2 + bx+ c$ and $g(x) = cx^2 + bx + a$ . If $|f(0)| \leq 1, |f(1)| \leq 1, |f(-1)| \leq 1$ , prove that for $|x| \leq 1$ ,(a) $|f(x)| \leq 5/4$ ,(b) $|g(x)| \leq 2$ .