 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/27k/b/solution

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

Q

gives

\partial_{y_{i}} e^{y^{T} Q y /2} = e^{y^{T} Q y /2} \sum_{j} q_{ij} y_{j}

. Also

\partial_{y_{i}} f (i, y^{2} /2) = y_{i} \partial_{ℓ_{i}} f

. Substitute these identities into the left side and integrate the resulting total

y_{i}

derivative by parts. Compact support removes the boundary term and leaves exactly the right side.

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