(a) Determine real quadratic functions $a(x), b(x), c(x)$ such that the interpolation formula,

$$f(x) \approx a(x) f(0)+b(x) f(2)+c(x) f(3),$$

is exact when $f(x)$ is any real polynomial of degree 2 .

(b) Use this formula to construct approximations for $f(5)$ and $f^{\prime}(1)$ which are exact when $f(x)$ is any real polynomial of degree 2 . Calculate these approximations for $f(x)=x^{3}$ and comment on your answers.

(c) State the Peano kernel theorem and define the Peano kernel $K(\theta)$. Use this theorem to find the minimum values of the constants $\alpha$ and $\beta$ such that

$$\left|f(1)-\frac{1}{3}[f(0)+3 f(2)-f(3)]\right| \leqslant \alpha \max _{\xi \in[0,3]}\left|f^{(2)}(\xi)\right|$$

and

$$\left|f(1)-\frac{1}{3}[f(0)+3 f(2)-f(3)]\right| \leqslant \beta\left\|f^{(2)}\right\|_{1}$$

where $f \in C^{2}[0,3]$. Check that these inequalities hold for $f(x)=x^{3}$.