Final answer:
The surface f(x, y) = 2 is a plane parallel to the XY plane. The surface is normal at all points and there is no specific direction for the greatest rate of increase of z as the function is not dependent on x or y.
Step-by-step explanation:
This question pertains to the field of calculus, specifically taking derivatives of multivariable functions and their concepts of normal vectors and rates of increase. The surface in the problem is given by f(x, y) = 2. A sketch would make this surface look like a plane parallel to the x-y plane sitting at z = 2. The surface intersects the z-axis at (0,0,2), and it does not intersect the x or y axes as it is parallel to the XY plane.
To find the normal to the surface at point P(1,2,0), we note that the gradient of a function gives the direction of maximum rate of increase. The gradient (∇f) is a vector pointing in the direction of the maximum rate of increase of z. For the given function, the gradient is always 0 as the function is not dependent on x and y. Therefore, not having a specific direction, the surface is considered normal at all points.
Finally, to determine in which direction to move for the greatest increase of z, we would also consider the direction of the gradient. However, since the rate of change of z is constant (z = 2), there isn't really a 'direction of greatest increase of z'. Regarding the second question whether this direction is the same as that points to the origin, the rate of change of z is constant everywhere so there's no unique direction pointing to the greatest rate of increase.
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