1. ## diff. calculus

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

ps. sorry for the mistake

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

3. Originally Posted by kuntah
oke thx man
Moderators should warn this kind of people.

4. 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.