Problem - 3625
Suppose $a, b, c$ are all real numbers. If the quadratic polynomial $P(x)=ax^2 + bx + c$ satisfies the condition that $\mid P(x)\mid \le 1$ for all $-1 \le x \le 1$, find the maximum possible value of $b$.
Because $b=(P(1)-P(-1))/2 \le 2/2 = 1$, the maximum possible value of $b$ is at most 1. Meanwhile, the polynomial $x^2/2 + x - 1/2$ satisfies the condition. Therefore we conclude the answer is $\boxed{1}$.