A linear functional acting on $f \in C^{k+1}[a, b]$ is approximated using a linear scheme $L(f)$. The approximation is exact when $f$ is a polynomial of degree $k$. The error is given by $\lambda(f)$. Starting from the Taylor formula for $f(x)$ with an integral remainder term, show that the error can be written in the form

$$\lambda(f)=\frac{1}{k !} \int_{a}^{b} K(\theta) f^{(k+1)}(\theta) d \theta$$

subject to a condition on $\lambda$ that you should specify. Give an expression for $K(\theta)$.

Find $c_{0}, c_{1}$ and $c_{3}$ such that the approximation scheme

$$f^{\prime \prime}(2) \approx c_{0} f(0)+c_{1} f(1)+c_{3} f(3)$$

is exact for all $f$ that are polynomials of degree 2 . Assuming $f \in C^{3}[0,3]$, apply the Peano kernel theorem to the error. Find and sketch $K(\theta)$ for $k=2$.

Find the minimum values for the constants $r$ and $s$ for which

$$|\lambda(f)| \leqslant r\left\|f^{(3)}\right\|_{1} \quad \text { and } \quad|\lambda(f)| \leqslant s\left\|f^{(3)}\right\|_{\infty}$$

and show explicitly that both error bounds hold for $f(x)=x^{3}$.