 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ii/paper-2/2f/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

The given moments imply

\int_{0}^{1} f (x) P (x) d x = 0

for every polynomial

P

P_{k}

converging uniformly to

f

\int_{0}^{1} f^{2} = \int_{0}^{1} f (f - P_{k}) \leq ∥ f - P_{k} ∥_{\infty} \int_{0}^{1} ∣ f ∣ ⟶ 0.

A nonnegative continuous function with zero integral vanishes everywhere. Hence $f = 0$ .

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