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 $| g (x) | \le 2$.
Problem
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist F1 day2 (F= Functions)
Tags: trinomial, algebra, inequalities