Consider $a,b,c\in\mathbb{R}$ such that $|ax^2+bx+c|\leq 1\;\forall x\in \left[0,1\right]$. Prove that $|a|\leq 8\;\;,|b| \leq 8$ and $|c| \leq 1$.
My Attempt:
Set $x = 0$ in $|ax^2+bx+c|\leq 1$ to get $|c| \leq 1$. Similarly set $x = 1$ in $|ax^2+bx+c| \leq 1$ to get $|a+b+c|\leq 1$.
From here, how can I calculate bounds for $a$ and $b$?