Lagrange Multipliers Using The Implicit Function Theorem

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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 ...