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Thread: Mean value theorem (multivariate case)

  1. #1
    Sep 2010

    Mean value theorem (multivariate case)

    If $\displaystyle f: \mathbb{R}^n \rightarrow \mathbb{R} $ is twice continuously differentiable, we have that:
    $\displaystyle \nabla f(x+p) = \nabla f(x) + \int_{0}^1 \nabla^2 f(x+tp) p dt$
    for some $\displaystyle t \in (0,1)$ and $\displaystyle p \in \mathbb{R}^n$

    I know it's related to the mean value theorem (multivariate case). Can anybody derive this result or give me a reference to where it is derived?

    Last edited by mr fantastic; Sep 8th 2010 at 12:57 PM. Reason: Re-titled.
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