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}
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