Bound on Coefficients

Question

For real $a,b,c$ the following holds $|ax^2+bx+c|\le 1 ; \forall x\in [0,1]$.Show that $|a|+|b|+|c|\le 17$. Cant show that the equality holds.I always get the lesser bounds.

Answer 1

Hint: If $P$ is your polynomial, you have $c=P(0)$, $\frac{a}{4}+\frac{b}{2}+c=P(1/2)$, $a+b+c=P(1)$. Compute $a,b,c$ in function of $P(0),P(1/2),P(1)$.

Answer 2

The maximum of $|ax^2+bx+c|\le 1$ in $[0,1]$ can be $x=0,1$ (border points of the interval $[0,1]$) or the critical point of $ax^2+bx+c$, namely $x=-b/2a$...