Let $f:\mathbb{R}^3\to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x, y_1, y_2)$. Assume that $f(3, -1, 2)=0$ and

Question

Let $f:\mathbb{R}^3\to \mathbb{R}^2$ be of class $C^1$; write $f$ in the form $f(x, y_1, y_2)$. Assume that $f(3, -1, 2)=0$ and $Df=\begin{bmatrix}1 & 2 & 1\\ 1& -1 & 1\end{bmatrix}$.

(a) Show there is a function $g : B\to \mathbb{R}^2$ of class $C^1$ defined on an open set $B$ in $\mathbb{R}$ such that $f(x, g_1(x) , g_2(x)) = 0$ for $x\in B$, and $g(3) = (-1, 2)$.

(b) Find $Dg(3)$.

(c) Discuss the problem of solving the equation $f(x,y_1 , y_2) = 0$ for an arbitrary pair of the unknowns in terms of the third, near the point $(3, -1, 2)$.

This exercise, is already posted here Application of Implicit Function theorem for this problem and here Implicit Function Theorem computation problem but there is no answer for (c), I would like to know how to solve (c), could someone help me please?

Answer 1

Basically you select the two columns in $Df$ you want to solve for in terms of the third variable. If the determinant is $0$, you are out of luck. Otherwise you can apply Implicit Function Theorem and repeat part $(a)$ and $(b)$