Here is a very old high school exam question I am trying to solve (purely for interest only):
If $a,b,c$ are real numbers such that $-1 \le ax^2+bx+c \le 1$ for $-1 \le x \le 1$ prove that $-4 \le 2ax+b \le4$ for $-1 \le x \le 1$ (Hint: Consider the functions at the end-points and at the mid-point of the interval).
I can see (graphically) that if $a>0$ then, as $2ax+b$ is the gradient function, the max and min gradients will occur when the parabola passes through the end-points $(-1,1)$ and $(1,1)$ and the mid-point $(0,-1)$. This gives 3 equations with 3 unknowns and is solved to give $a=2, b=0$ and $c=-1$. The required result follows easily from this. Due to symmetry, a<0 gives the same result.
Can someone please help turn my "partial" solution into a more convincing/algebraic solution.