how to prove that the sum of the power two of three reals is less than 5? $a, b$ and $c$ are three real number such that: $(\forall x \in [-1,1]): |ax^2+bx+c|\leq 1$ .
Prove that:
1)   $|c|\leq 1$ .
2) $-1\leq a+c\leq 1$ .
3)Deduce that : $a^2+b^2+c^2\leq 5$.
The first and second questions are easy to prove ( just take $x=0, -1$ or $1$ ... ), but the third one ?!!! I'm sure that the responce must be easy since it's a deduction. 
I need your help to solve it, thank you.

I'm not sure how you're supposed to go straight from 1) and 2) to 3) without some additional steps.
The help I would offer is that it might help to strengthen 2) to the statement $|a+c| + |b| \le 1$ and work from there.
