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Lagrange Multipliers Using The Implicit Function Theorem

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Please refer to the attachments for the questions.

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Solution Summary

We clarify the use of the Implicit Function Theorem (IFT) in the proof of the theorem of Lagrange multipliers. We define the functions and check the hypotheses to use the IFT. We then check that the consequences provided by the IFT give what we need to prove the theorem of Lagrange multipliers.

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Please see the attached file for a properly formatted answer.

There is a bit of confusion when it comes to notation. Theorem 3 and the Inverse function Theorem as stated in the file you provided shared some symbols such as $mathbf{X}, mathbf{U}$ but they are used differently as we point out below. To make things more clear I will use $mathbf{Y}$ instead of $mathbf{U}$ to denote $(x_3,dotsc,x_n)$. I write inside parentheses other comments about notation.

Using the notation in Theorem 3 and its proof given in the attachment (except for $mathbf{Y}$ instead of $mathbf{U}$) let us write the problem in the setting of the Implicit Function Theorem (IFT) as stated in the attached file.

We write $R^n=R^{(n-2)+2}$ (in IFT we use $n$ ``equals'' $n-2$ and $m=2$) and a point
$(x_1,x_2,x_3,dotsc,x_n)=(mathbf{V},mathbf{Y})$, where $mathbf{V}=(x_1,x_2)inR^2$ and
$mathbf{Y}=(x_3,dotsc,x_n)inR^{n-2}$ (we are using ...

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