Solve system of the two equations:
2x - 6 +2y=0
4y - 10 +2x=0
Why this will give you your answer?
STATEMENT: Find all points (x,y) at which the tangent plane to the graph of z = x^2 - 6x + 2y^2 -10y + 2xy is horizontal.
ANSWER: The answer in the back of the book is (1,2), but I have no idea how you get that answer.
ATTEMPT: I think to start the problem you need to find the gradient vector and then use the point (0,0,1) because it's the unit vector in the z direction where it's perpendicular to the horizontal plane. That's as far as I gotten and I do not know where to go from here or if the way I started the problem is even right.
(DEL)(which is an upside down triangle) f (x,y) = (2x - 6 +2y)i + (4y - 10 +2x)j
But what I would do is define so that and then argue that the tangent plane will be parallel to the xy-plane if and only if that vector is in the z-direction. That is, that 2x- 6+ 2y= 0 and 4y- 10+ 2x= 0 as Also sprach Zarathustra said.