Determine the real coefficients $b_{1}, b_{2}, b_{3}$ such that

$$\int_{-2}^{2} f(x) d x=b_{1} f(-1)+b_{2} f(0)+b_{3} f(1)$$

is exact when $f(x)$ is any real polynomial of degree 2 . Check explicitly that the quadrature is exact for $f(x)=x^{2}$ with these coefficients.

State the Peano kernel theorem and define the Peano kernel $K(\theta)$. Use this theorem to show that if $f \in C^{3}[-2,2]$, and $b_{1}, b_{2}, b_{3}$ are chosen as above, then

$$\left|\int_{-2}^{2} f(x) d x-b_{1} f(-1)-b_{2} f(0)-b_{3} f(1)\right| \leqslant \frac{4}{9} \max _{\xi \in[-2,2]}\left|f^{(3)}(\xi)\right|$$