Max Magnitude of a,b,c in Quaratic Equation |ax^2+bx+c|<=1

[ritwik06]

Homework Statement

$$|ax^{2}+bx+c|\leq1 \forall x\in[0,1]$$
Find the maximum magnitude of a,b, and c possible!

My attempt:
putting x=0;
|c|<=1
hence its maximummagnitude is 1.
Help me with a and b.
regards,
Ritwik

 

[dynamicsolo]

You aren't going to be able to set limits on the coefficients separately, I think. You are probably going to have relationships between the coefficients.

Picture it this way. The inequality is saying that

$$-1 \leq ax^{2}+bx+c \leq 1$$ for $$0 \leq x \leq 1$$.

So a portion of the curve for this function has to fit entirely in the "box" bounded by x = 0, x = 1, y = -1, and y = 1. (Obviously, the parabola goes on forever elsewhere...) What does that mean for intersection points of the parabola with the "box"? How might you find coefficients from that?