Suppose $p(x)=ax^2+bx+c$ is a quadratic polynomial with real coefficients and $|p(x)| \leq 1$ for all values of $x$ in the range $[0,1]$. Prove that maximum possible value of $|a|+|b|+|c|$ is $17$.
I could not even start the problem. Any hints would be helpful.
Hint: $$a=2f(1)+2f(0)-4f(\frac{1}{2}),b=4f(\dfrac{1}{2})-f(1)-3f(0)，c=f(0)$$ so $$|a|le 2|f(1)|+4|f(\frac{1}{2})|+2|f(0)|=2+4+2=8$$ simaler $|b|\le 8,|c|\le 1$.then we have $$|a|+|b|+|c|\le 17$$