An implicit function of two variables
Task number: 3222
For the given relation \(x^2 + 2y^2 + 3z^2 + xy - z - 9 = 0\).
Variant 1
Prove that this relation defines a smooth function \(z = z(x, y)\), in some neighborhood \(U\) of the point \([1, -2]\) satisfying \(z(1, -2) = 1\).
Variant 2
Determine \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) in the neighborhood \(U\).
Variant 3
Write down the equation of the plane tangent to the function \(z\) at the point \([1, -2]\).
