MathDB

trinomial inequalities

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist F1 day2 (F= Functions)

September 24, 2021

trinomialalgebrainequalities

Problem Statement

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