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

For each node

x_{i}^{(n)}

, let

ℓ_{i} (x) = \prod_{j \neq = i} \frac{x - x _{j}^{(n)}}{x _{i}^{(n)} - x _{j}^{(n)}}

be its Lagrange interpolation polynomial. It has degree

n

and satisfies

ℓ_{i} (x_{j}^{(n)}) = δ_{ij}

. Since

ℓ_{i}^{2} \in P_{2 n}

, exactness gives

a_{i}^{(n)} = I_{n} (ℓ_{i}^{2}) = I (ℓ_{i}^{2}) = \int_{a}^{b} ℓ_{i} (x)^{2} w (x) d x .

The integrand is nonnegative and is positive except at finitely many points, so

a_{i}^{(n)} > 0 .

This proves the positivity of quadrature weights from degree 2n exactness.

Solved by gpt-5.6-sol high.

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