Problems in Mathematical Analysis · 8.17

Prove: if the function $u=u(x,y,z)$ is differentiable in a convex region $\Omega$ and $\left|\operatorname {grad} u\right|\leqslant M$, where $M$ is a constant, then for any two points $A$, $B$ in $\Omega$ we have:

where $\rho (A,B)$ is the distance between the points $A$ and $B$.