Notes:Mean value theorem

Overview

There are at least 2 forms of the mean value theorem - a first form, encountered by first years for single variables, then a multivariable one, THEN one for smooth manifolds. This page documents them, so I can separate them into 1 or 2 or 3 forms and then document them on the root page Mean value theorem.

Statements

Maurin

For $f:[a,b]\rightarrow\mathbb{R}$

• Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function, differentiable on $(a,b)\subset[a,b]$, then:
  • $\exists c\in(a,b)$ such that $$f'(c)=\frac{f(a)-f(b)}{b-a}$$
• I suspect this can be generalised to: $f$ continuous on a closed connected set, and differentiable on the interior, then the result follows.

For $f:[x,x+h]\subseteq U\rightarrow\mathbb{R}$ for $U$ open in a Banach space

He first notes: $[x,x+h]:=\{y\in X\ \vert\ y=(1-a)x+a(x+h),\ a\in[0,1]\}$, this just says $y$ goes from $x$ to $x+h$ as $a$ goes from $0$ to $1$. A line in $(X,\Vert\cdot\Vert)$.

• Given $f:U\rightarrow\mathbb{R}$ for an open set, $U$ in a Banach space $(X,\Vert\cdot\Vert)$, if $f$ is differentiable at every point of the interval $[x,x+h]$ For what $h$? then:
  • $\exists\theta\in(0,1)$ such that:
    • $$f(x+h)-f(x)=f'(x+\theta h)\cdot h$$

For $f:U\rightarrow Y$ for $U$ open in $(X,\Vert\cdot\Vert_X)$ mapping to $(Y,\Vert\cdot\Vert_Y)$

• If $f$ has continuous derivative at every point of the interval $[x,x+h]$ then:
  • $$\Vert f(x+h)-f(x)\Vert_Y\le \Vert h\Vert_X\cdot \mathop{\text{Sup} }_{\theta\in[0,1]}\left(\Vert f'(x+\theta h)\Vert_?\right)$$ What is $\Vert\cdot\Vert_?$ actually defined on?

Topology and Geometry

For $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$

• Let $f:\mathbb{R}^n\rightarrow\mathbb{R} \in$ $C^1$ - the class of functions with continuous partial derivatives. Let $x=(x_1,\ldots,x_n)$ and $\bar{x}=(\bar{x_1},\ldots,\bar{x_n})$. Then:
  • $$f(x)-f(\bar{x})=\sum^n_{i=1}\frac{\partial f}{\partial x_i}(\tilde{x})(x_i-\bar{x_i})$$ for some $\tilde{x}$ on the line segment between $x$ and $\bar{x}$

Corollary

• Let $f:\mathbb{R}^k\times\mathbb{R}^m\rightarrow\mathbb{R}$ be $C^1$. For $x\in\mathbb{R}^k$ and $y\in\mathbb{R}^m$ then:
  • $f(x,y)-f(x,\bar{y})=\sum^m_{i=1}\frac{\partial f}{\partial y_i}(x,\tilde{y})(y_i-\bar{y_i})$ for some $\tilde{y}$ on the line segment between $y$ and $\bar{y}$