CR-MO 2016 F1

Let $a, b$ and $c$ be real numbers, and let $f (x) = ax^2 + bx + c$ and $g (x) = cx^2 + bx + a$ functions such that $| f (-1) | \le 1$ , $| f (0) | \le 1$ and $| f (1) | \le 1$ . Show that if $-1 \le x \le 1$ , then $| f (x) | \le \frac54$ and $| g (x) | \le 2$ .