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

Suppose that an $(n+1)$-node quadrature rule were exact on ${\mathcal{P}}_{2n+2}$. The nodal polynomial $Q_{n+1}$ has degree $n+1$, so $Q_{n+1}^2\in {\mathcal{P}}_{2n+2}$. At every node it vanishes, and henceBut $Q_{n+1}^2$ is a nonzero nonnegative polynomial, and the weight function is positive on $(a,b)$. Thereforea contradiction. This is the degree ceiling for quadrature exactness.