Problem

Source: China Southeast Mathematical Olympiad

Tags: inequalities



Let $a, b, c$ be real numbers, $a \neq 0$. If the equation $2ax^2 + bx + c = 0$ has real root on the interval $[-1, 1]$. Prove that $$\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},$$and determine the necessary and sufficient conditions of $a, b, c$ for the equality case to be achieved.