Let $c$ be a real number. If the inequality $$f(c)\cdot f(-c)\ge f(a)$$holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.
Source: 2022 Turkey JBMO TST P6
Tags: Inequality, real number
Let $c$ be a real number. If the inequality $$f(c)\cdot f(-c)\ge f(a)$$holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.