The answer is yes. Indeed, this follows from the case when $n=1$, because for each $x\in\mathbb R^n$ we have $M(f)(x):=\max\{\|h\|\colon h\in\partial f(x)\}=\max\{M(f|_\ell)(x)\colon\ell\in L_x\}$, where $L_x$ is the set of all straight lines through $x$ and $f|_\ell$ is the restriction of $f$ to $\ell$.