User:Keith.U/Mean Value Theorem (Euclidean Space)

Theorem
Let $A \subseteq \R^{n}$ be a region.

Let $a, b \in A$.

Let $L$ denote the line segment joining $a$ to $b$.

Let $f : A \to \R$ be differentiable on $L$.

Then:
 * $\exists c \in L : f \left({ b }\right) - f \left({ a }\right) = \nabla f \left({ c }\right) \left({ b - a }\right)$

where $\nabla f$ denotes the gradient of $f$.