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For any $x,y \in S$ we have $$f(y)=f(x) + \nabla f(x)^T(y-x) + \frac{1}{2}(y-x)^T\nabla^2f(z)(y-x) $$

for some $z$ on the line segment $[x,y].$

I don't see why this should be true, other than that it 'feels' right. Is there a simple proof? Also, I'm not sure if the title is most appropriate, so feel free to edit it.

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I'm assuming of course that the function is sufficiently smooth for second derivative to exist and be continuous. –  Squirtle Jul 10 '13 at 0:05
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This is just Taylor's theorem, with the remainder in a certain form. You can derive Taylor's theorem easily using integration by parts. Then use the mean value theorem for integration to get the remainder in the desired form. See math.stackexchange.com/a/315412/35914 –  wj32 Jul 10 '13 at 0:10
    
Cool... thanks! –  Squirtle Jul 10 '13 at 20:10
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