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

be a formula of numerical differentiation which is exact on polynomials of degree 2 , and let

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

be its error.

Find the values of the coefficients $a_{0}, a_{1}, a_{2}$.

Using the Peano kernel theorem, find the least constant $c$ such that, for all functions $f \in C^{3}[0,2]$, we have

$$|e(f)| \leqslant c\left\|f^{\prime \prime \prime}\right\|_{\infty} .$$