Has anybody got an idea how the equations
$\max\limits_{\partial B_r(x)}u = u\left(x+r \frac{Du(x)}{|Du(x)|}\right)$
$\min\limits_{\partial B_r(x)}u = u\left(x-r \frac{Du(x)}{|Du(x)|}\right)$
arise for linear and non-constant $u:\mathbb{R}^n\to\mathbb{R}$? It seems heuristically correct but I can't find a way to prove it. I tried to play with the linearity of $u$ around a little bit but I couldn't get near those equations unfortunately.
As $u$ is linear, it's gradient $Du = d$ is constant, as pointed by @PierreCarre. You can then write $$u(x + h) = u(x) + \langle Du(x), h \rangle = u(x) + \langle d, h \rangle. $$ You can then use the Cauchy-Schwarz inegality to demonstrate you result.