Question: Consider the system of equations

\[\left\{\begin{array}{lrl} x_{1} y_{2}^{2}-2 x_{2} y_{3} & = & 1 \\ x_{1} y_{1}^{5}+x_{2} y_{2}-4 y_{2} y_{3} & = & -9 \\ x_{2} y_{1}+3 x_{1} y_{3}^{2} & = & 12 \end{array}\right.\]

1. Show that there is a continuously differentiable function \(\phi\) expressing \(y_{1}, y_{2}, y_{3}\) in terms of \(x_{1}, x_{2}\) in a neighbourhood of \(\left(x_{1}, x_{2}, y_{1}, y_{2}, y_{3}\right)=(1,0,-1,1,2)\). That is, show that there exists an open set \(V \ni(1,0,-1,1,2)\), an open set \(U \ni(1,0)\) and \(\phi: U \rightarrow \mathbb{R}^{3}\) such that \(\phi(1,0)=(-1,1,2)\) and \(\left(x_{1}, x_{2}, y_{1}, y_{2}, y_{3}\right)\) is in \(V\) and satisfies all three equations above iff \(\left(x_{1}, x_{2}\right) \in U\) and \(\left(y_{1}, y_{2}, y_{3}\right)=\) \(\phi\left(x_{1}, x_{2}\right)\)
2. Evaluate \(\frac{\partial \phi_{1}}{\partial x_{1}}, \frac{\partial \phi_{2}}{\partial x_{1}}, \frac{\partial \phi_{3}}{\partial x_{1}}\) at \((1,0)\).

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