Let

$$f^{\prime \prime}(0) \approx a_{0} f(-1)+a_{1} f(0)+a_{2} f(1)=\mu(f)$$

be an approximation of the second derivative which is exact for $f \in \mathcal{P}_{2}$, the set of polynomials of degree $\leq 2$, and let

$$e(f)=f^{\prime \prime}(0)-\mu(f)$$

be its error.

(a) Determine the coefficients $a_{0}, a_{1}, a_{2}$.

(b) Using the Peano kernel theorem prove that, for $f \in C^{3}[-1,1]$, the set of threetimes continuously differentiable functions, the error satisfies the inequality

$$|e(f)| \leq \frac{1}{3} \max _{x \in[-1,1]}\left|f^{\prime \prime \prime}(x)\right| .$$