Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 2 / b / Solution

A two-node collocation method already has order at least two. It has order at least three exactly when its node polynomial is orthogonal to constants on $[0,1]$:Equivalently,This is the additional quadrature order condition ${\sum }_i{b}_i{c}_i^2=1/3$.