[Steps Shown] The point P=(-2,1,1) satisfies the equation z^3+x y^2 z+1=0. Can we implicitly define z as a function of x and y using this equation in a region
Question: The point \(P=(-2,1,1)\) satisfies the equation \(z^{3}+x y^{2} z+1=0\). Can we implicitly define \(z\) as a function of \(x\) and \(y\) using this equation in a region about $P ?$ Compute \(\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}\), and \(\frac{\partial^{2} z}{\partial x^{2}}\) if it makes sense to do so.
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