Like you said, and , which is a system of equations. Solutions to the system of equations are the points on the surface at which the tangent plane is horizontal.
This is a question from my calculus 3 class.
"Find all points on the surface at which the tangent plane is horizontal."
z = x^2 - xy + y^2 - 2x + 4y
Okay, so I'm not sure where to begin. I know that the partial derivatives z_x and z_y are both zero. And so I've thought of making a tangent equation but I got stuck.
Tangent plane = (z - z0) ?
I'm not sure if I have to solve for x and y in the given equation, or really what I have to do in general.
Can anyone give me a push in the right direction? Maybe I'm over-thinking this problem.