Mean Value Theorem

Theorem multiple variables

• $G \subseteq \mathbb{R}^{n}$,
  • convex
  • open
• $f:G \rightarrow \mathbb{R}$,
  • differentiable
• $x, y \in G$,

\[\exists \in [0, 1] \text{ s.t. } f(y) - f(x) = \nabla f(x + c(y - x))^{\mathrm{T}} (y - x) .\]

proof

Let be define

\[g(t) := f(x + t(y - x)) .\]

Since $g$ is differentiable, by mean value theorem for one dimentional function,

\[\exists c \in [0, 1] \text{ s.t. } g(1) - g(0) = g^{\prime}(c) .\]

Hence

\[f(y) - f(x) = \nabla f(x + c(y - x))^{\mathrm{T}} (y - x) .\]

$\Box$

Reference