Solution
= Solution
If the bound is to hold for all $f\in C^3[-1,1]$, the error functional must annihilate every polynomial of degree at most two, because such a polynomial has $f'''=0$. Applying it to $1,x,x^2$ gives
$$
\begin{aligned}
a_{-1}+a_0+a_1&=0,\\
-a_{-1}+a_1&=0,\\
a_{-1}+a_1&=2.
\end{aligned}
$$
Therefore
$$
\boxed{a_{-1}=1,\qquad a_0=-2,\qquad a_1=1},
$$
so $\eta(f)=f(-1)-2f(0)+f(1)$ is the <central finite difference> approximation.