Starting from the Taylor formula for $f(x) \in C^{k+1}[a, b]$ with an integral remainder term, show that the error of an approximant $L(f)$ can be written in the form (Peano kernel theorem)

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

when $L(f)$, which is identically zero if $f(x)$ is a polynomial of degree $k$, satisfies conditions that you should specify. Give an expression for $K(\theta)$.

Hence determine the minimum value of $c$ in the inequality

$$|L(f)| \leq c\left\|f^{\prime \prime \prime}\right\|_{\infty}$$

when

$$L(f)=f^{\prime}(1)-\frac{1}{2}(f(2)-f(0)) \text { for } f(x) \in C^{3}[0,2]$$