Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 17C / a / 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 $f^{\prime \prime \prime }=0$. Applying it to $1,x,x^2$ givesThereforeso $\eta (f)=f(-1)-2f(0)+f(1)$ is the central finite difference approximation.