If $f(x)=ax^2+bx+c$ be such that $|f(0)|\leq 1$,$|f(1)\leq 1$ and $|f(-1)\leq 1$, then for $x\in [-1,1]$ then what may be maximum value of $|f(x)|$

Question

If $f(x)=ax^2+bx+c$ be such that $|f(0)|\leq 1$,$|f(1)|\leq 1$ and $|f(-1)|\leq 1$, then for $|x|\leq 1$, $|f(x)|$ cannot have the value

(A) $\frac{1}{4}$

(B) 1

(C) $\frac{5}{4}$

(D) $\frac{7}{4}$

My Attempt

I am trying to fix $x=-\frac{b}{2a}$ between $-1$ and $1$ so that all conditions may get satisfied but am not able to reach any conclusion