Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 17B / c / Solution

For each node $x_i^{(n)}$, letbe its Lagrange interpolation polynomial. It has degree $n$ and satisfies ${\ell }_i(x_j^{(n)})={\delta }_{ij}$. Since ${\ell }_i^2\in {\mathcal{P}}_{2n}$, exactness givesThe integrand is nonnegative and is positive except at finitely many points, soThis proves the positivity of quadrature weights from degree 2n exactness.