 Solution

ID: past-exam-of-the-options-i-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

y

, then for

x \neq = y

there exists

s \in (0, 1)

such that

f (y) - f (x) = D f (x + s (y - x)) (y - x) = \nabla f (x + s (y - x)) \cdot (y - x) .

This follows by applying the one-dimensional mean value theorem to

t \mapsto f (x + t (y - x))

[0, 1]

. The segment condition is necessary for this formulation on a possibly nonconvex domain.

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