Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / F / 1 / b / i / Solution

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.