Find a relation between $a$ and $b$?

Question

I would appreciate if somebody could help me with the following problem:

Let $f(x)=x^2-2ax+b$, $a,b\in \mathbb{R}$

Q: Find a relation between $a$ and $b$ ?

If $|x|\leq 1$ then $|f(x)|\leq1 $

Answer 1

It may be helpful for you.

$x^2 - 2ax + b = (x - a)^2 + (b - a^2)$

As $f(x) \le 1$ we shall get $(x - a)^2 + (b - a^2) \le 1$.

Also $|x| \le 1$, after a few steps of calculation we shall get $(x - a)^2 \ge (1 + a)^2$.

Thus $(b - a^2) \le 1 - (x - a)^2 \le 1 - (1 + a)^2$.

Now simplify.