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

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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 $f$.