Past exam of the options I course of the University of Oxford / 2013 / F / 1 / b

The scalar-valued multivariable mean value theorem states that, if $f$ is differentiable on an open set containing the closed line segment from $x$ to $y$, then for $x\ne y$ there exists $s\in (0,1)$ such thatThis follows by applying the one-dimensional mean value theorem to $t\mapsto f(x+t(y-x))$ on $[0,1]$. The segment condition is necessary for this formulation on a possibly nonconvex domain.

For every $z\in B$, $z_3+3>2$ and $|z_1+z_2|<2$. Consequently each of the three partial derivatives in the preceding gradient has absolute value below $1/2$, soThe segment joining $x,y\in B$ lies inside $B$. The mean value theorem and Cauchy-Schwarz inequality therefore giveFor $x=y$ both sides vanish. This proves the required Lipschitz continuity uniformly on the ball.