diff. calculus

**kuntah** · November 12th 2007, 02:44 PM

How can I prove the n dim analogue of the m value theorem?

ps. sorry for the mistake

**ThePerfectHacker** · November 12th 2007, 05:01 PM

Let $D\subseteq \mathbb{R}^n$ . Say that $\bold{a},\bold{b}\in D$ and the line segment between $\bold{a},\bold{b}$ is contained in $D$ i.e. $\bold{a}+t(\bold{b}-\bold{a}) \in D \mbox{ for }t\in [0,1]$ . Now define $g(t) = f(\bold{a}+t(\bold{b}-\bold{a})$ and confirm that $g$ is a differentiable function on $(0,1)$ (by multi-variable chain rule) and continous on $[0,1]$ so by the basic mean value theorem we have $g'(c)= g(1) - g(0)$ but $g'(c) = \nabla f(\bold{c})\cdot (\bold{b}-\bold{a})$ . That means, $\nabla f(\bold{c}) \cdot (\bold{b}-\bold{a}) = f(\bold{b})-f(\bold{a})$ for some $\bold{c}$ between the line segment. Q.E.D.

**Krizalid** · November 13th 2007, 08:22 AM

Originally Posted by kuntah

oke thx man

Moderators should warn this kind of people.

**ThePerfectHacker** · November 13th 2007, 09:46 AM

Originally Posted by Krizalid

Moderators should warn this kind of people.

Talk appropriately. And do not tell me how I should do my job.

I give enough warnings.

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