Lately, we've been going over these two theorems in class. I have a few questions to put forth.(adsbygoogle = window.adsbygoogle || []).push({});

1) I know that in lower spaces, an inverse of a function exists locally (say around a point G) if it does not attain it's max/min at G (i.e. if f'(G) doesn't equal 0). Now, with the inverse function theorem, we have that det| Df | doesn't equal 0 (where Df is the matrix of partial derivatives?). I'm unsure how the determinant of the matrix relates to max/min at a point- So i can relate the two ideas.

2)I know that the inverse function theorem implies the implicit function theorem, and have seen proofs of it, but how can we prove that the implicit function theorem implies the inverse function theorem? (any suggestions on how to go about this proof would be appreciated)

3)In the implicit function theorem, where we consider a vector in R^m as [x y], x in R^(m-n), y in R^n, why is it that we only need the the determinant of DF/DY to not equal 0? Is this where the inverse function theorem comes in (to create a new function G s.t. G(x)=y?). What exactly is that matrix DF/DY? Also, why does there need exist a solution to the function?

4) In both theorems, why is it required to have f continuously differentiable?

Sorry if I'm asking too many questions, as I know some may flame me for this fact, but thanks in advance as any help is very appreciated.

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# Implicit and inverse function theorem

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