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 [ilmath]f:[a,b]\rightarrow\mathbb{R} [/ilmath]

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

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

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

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

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

• If [ilmath]f[/ilmath] has continuous derivative at every point of the interval [ilmath][x,x+h][/ilmath] then:
  • [math]\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)[/math] What is [ilmath]\Vert\cdot\Vert_?[/ilmath] actually defined on?

Topology and Geometry

For [ilmath]f:\mathbb{R}^n\rightarrow\mathbb{R}^n[/ilmath]

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

Corollary

• Let [ilmath]f:\mathbb{R}^k\times\mathbb{R}^m\rightarrow\mathbb{R} [/ilmath] be [ilmath]C^1[/ilmath]. For [ilmath]x\in\mathbb{R}^k[/ilmath] and [ilmath]y\in\mathbb{R}^m[/ilmath] then:
  • [ilmath]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})[/ilmath] for some [ilmath]\tilde{y} [/ilmath] on the line segment between [ilmath]y[/ilmath] and [ilmath]\bar{y} [/ilmath]