Starting from Taylor's theorem with integral form of the remainder, prove the Peano kernel theorem: the error of an approximant $L(f)$ applied to $f(x) \in C^{k+1}[a, b]$ can be written in the form

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

You should specify the form of $K(\theta)$. Here it is assumed that $L(f)$ is identically zero when $f(x)$ is a polynomial of degree $k$. State any other necessary conditions.

Setting $a=0$ and $b=2$, find $K(\theta)$ and show that it is negative for $0<\theta<2$ when

$$L(f)=\int_{0}^{2} f(x) d x-\frac{1}{3}(f(0)+4 f(1)+f(2)) \quad \text { for } \quad f(x) \in C^{4}[0,2] .$$

Hence determine the minimum value of $\rho$ for which

$$|L(f)| \leqslant \rho\left\|f^{(4)}\right\|_{\infty}$$

holds for all $f(x) \in C^{4}[0,2]$.