 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-3/11a/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

m \leq ϕ (x) \leq M

on the compact sphere. Integrate these inequalities over its area

4 π r^{2}

4 π r^{2} m \leq \int_{\partial V_{r}} ϕ d S \leq 4 π r^{2} M, m \leq ϕ (0) \leq M .

The centre value is an average with positive weights, so it lies between the smallest and largest boundary-sphere values.

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