Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation $$-1\leq f(x)\leq 1 $$ for all $-1\leq x\leq 1$, then the maximum value of $f'(x)$ is
I think the maximum value of $f'$ is to be found within the limits. I do not have any idea how to do it. I tried analyzing with the help of the graph, the part of the curve must be well inside the square with vertices $(1,1);(-1,1);(-1,-1),(1,-1)$. But I failed. The answer is an integer between 0 and 9. Please give an elegant solution.