 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/6c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

ℓ_{i}

be the degree-

n - 1

cardinal polynomial with

ℓ_{i} (c_{j}) = δ_{ij}

. Exactness applied to the nonnegative polynomial

ℓ_{i}^{2}

gives

b_{i} = \sum_{j} b_{j} ℓ_{i} (c_{j})^{2} = \int_{a}^{b} w (x) ℓ_{i} (x)^{2} d x > 0,

because

w > 0

and

ℓ_{i}

is nonzero. Thus every Gaussian quadrature weight is positive.

 New to topics? Read the docs here!