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:
 * $\exists c \in L : \map f b - \map f a = \nabla \map f c \paren {b - a}$

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