Differentiate the ball mean-value formula and use the <divergence theorem>:
$$
Du(x_0)=\frac1{|B_r|}\int_{B_r(x_0)}Du
=\frac1{|B_r|}\int_{\partial B_r(x_0)}u\nu.
$$
Consequently
$$
|Du(x_0)|\leq\frac{|\partial B_r|}{|B_r|}\sup_{B_r(x_0)}|u|
=\frac nr\sup_{B_r(x_0)}|u|.
$$
Thus one may take $C(n)=n$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-107/1/b/solution