Corollaries: The MVT for Differentiable Functions from Rn to Rm

# Corollaries to the Mean Value Theorem for Differentiable Functions from Rn to Rm

Recall from The Mean Value Theorem for Differentiable Functions from Rn to Rm page that if $S \subseteq \mathbb{R}^n$ is open and $\mathbf{f} : S \to \mathbb{R}^m$ is differentiable on $S$ then if $\mathbf{x}, \mathbf{y} \in S$ are such that the line segment connecting $\mathbf{x}$ and $\mathbf{y}$ denoted $L(\mathbf{x}, \mathbf{y}) = \{ (1 - t)\mathbf{x} + t\mathbf{y} : t \in [0, 1] \}$ is contained in $S$, i.e., $L(\mathbf{x}, \mathbf{y}) \subset S$, then for every $\mathbf{a} \in \mathbb{R}^m$ there exists a point $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:

(1)
\begin{align} \quad \mathbf{a} \cdot [\mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x})] = \mathbf{a} \cdot [\mathbf{f}'(\mathbf{z}) (\mathbf{y} - \mathbf{x})] \end{align}

We will now look at some nice corollaries from this generalized Mean Value Theorem.

 Corollary 1 (The Multivariable Real-Valued Mean Value Theorem): If $S \subseteq \mathbb{R}^n$ is open and $f : S \to \mathbb{R}$ is differentiable on $S$ then for any two points $\mathbf{x}, \mathbf{y} \in S$ with $L(\mathbf{x}, \mathbf{y}) \subset S$ there exists a $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that $f(\mathbf{y}) - f(\mathbf{x}) = \nabla f(\mathbf{z}) \cdot (\mathbf{y} - \mathbf{z})$.
• Proof: By the Mean Value Theorem, for all $a \in \mathbb{R}$ there exists a $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:
(2)
\begin{align} \quad a [ f(\mathbf{y}) - f(\mathbf{x})] = a[f'(\mathbf{z})(\mathbf{y} - \mathbf{x})] \end{align}
• Set $a = 1$. Then:
(3)
\begin{align} \quad f(\mathbf{y}) - f(\mathbf{x}) = f'(\mathbf{z})(\mathbf{y} - \mathbf{x}) \end{align}
• Now the total derivative of $f$ at $\mathbf{z}$ is the Jacobian of $f$ at $\mathbf{z}$. But the Jacobian of $f$ at $\mathbf{z}$ is simply the gradient of $f$ at $\mathbf{z}$, i.e., $f'(\mathbf{z}) = \nabla f(\mathbf{z})$. Substituting this into the equation above gives us:
(4)
\begin{align} \quad f(\mathbf{y}) - f(\mathbf{x}) = \nabla f(\mathbf{z}) \cdot (\mathbf{y} - \mathbf{x}) \quad \blacksquare \end{align}
 Corollary 2: If $S \subseteq \mathbb{R}^n$ is open and $\mathbf{f} : S \to \mathbb{R}^m$ is differentiable on $S$ then for any two points $\mathbf{x}, \mathbf{y} \in S$ with $L(\mathbf{x}, \mathbf{y}) \subset S$ there exists a point $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that $\| \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) \| \leq M \| \mathbf{y} - \mathbf{x} \|$ where $\displaystyle{M = \sum_{k=1}^{m} \| \nabla f_k(\mathbf{z}) \|}$.

We write $\mathbf{f} = (f_1, f_2, ..., f_m)$.

• Proof: By the Mean Value Theorem, for any $\mathbf{a} \in \mathbb{R}^m$ such that $\| \mathbf{a} \| = 1$ there exists a $\mathbf{z} \in L(\mathbf{x}, \mathbf{y})$ such that:
(5)
\begin{align} \quad \mathbf{a} \cdot [\mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x})] = \mathbf{a} \cdot [\mathbf{f}'(\mathbf{z})(\mathbf{y} - \mathbf{x})] \end{align}
• Take the norm of both sides to get:
(6)
\begin{align} \quad \| \mathbf{a} \cdot [\mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x})] \| &= \| \mathbf{a} \cdot [\mathbf{f}'(\mathbf{z})(\mathbf{y} - \mathbf{x})] \| \\ \| \mathbf{a} \| \| \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) \| &= \| \mathbf{a} \| \| \mathbf{f}'(\mathbf{z})(\mathbf{y} - \mathbf{x}) \| \\ \| \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) \| &= \| \mathbf{f}'(\mathbf{z})(\mathbf{y} - \mathbf{x}) \| \\ \end{align}
(7)
\begin{align} \quad \| \mathbf{f}(\mathbf{y}) - \mathbf{f}(\mathbf{x}) \| & \leq M \| \mathbf{y} - \mathbf{x} \| \quad \blacksquare \end{align}