2004-41.md

course	course_year	question_number	tags	title	year
Numerical Analysis	IB	41	IB 2004 Numerical Analysis	3.II.22A	2004

Given $f \in C^{3}[0,1]$, we approximate $f^{\prime}\left(\frac{1}{3}\right)$ by the linear combination

$$\mathcal{T}[f]=-\frac{5}{3} f(0)+\frac{4}{3} f\left(\frac{1}{2}\right)+\frac{1}{3} f(1)$$

By finding the Peano kernel, determine the least constant $c$ such that

$$\left|\mathcal{T}[f]-f^{\prime}\left(\frac{1}{3}\right)\right| \leq c\left|f^{\prime \prime \prime}\right|_{\infty} .$$