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Math Help - diff. calculus

  1. #1
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    diff. calculus

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

    ps. sorry for the mistake
    Last edited by kuntah; November 13th 2007 at 12:34 PM. Reason: the warning
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    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.
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  3. #3
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    Quote Originally Posted by kuntah View Post
    oke thx man
    Moderators should warn this kind of people.
    Last edited by ThePerfectHacker; November 13th 2007 at 09:46 AM.
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  4. #4
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    Quote Originally Posted by Krizalid View Post
    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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