Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 17C (Numerical Analysis)

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.

The Peano kernel theorem says that if a continuous linear functional $L$ annihilates polynomials of degree below $m$, thenHere $m=3$ andApplying $L$ to $(x-t{)}_{+}^2/2$ givesHenceThe smallest candidate is therefore

The constant is sharp because continuous functions bounded by one can approximate $\mathrm{sgn}K$ arbitrarily closely outside an arbitrarily small interval around zero. Choose $f^{\prime \prime \prime }$ to be such an approximation and integrate three times to obtain $f\in C^3[-1,1]$. ThenNo smaller constant can therefore satisfy the inequality for every $f$.