as you say.
Yes that is correct.(ii) the maximum rate of change of f at P is just the length of its gradient vector at this point, ie as per (i).
You need to find another point on that surface so that , , and . The last is always true. Solve the first two for x and y. There will be two solutions, of course, one of which is x= 2, y= 1. Check that the other solution also satisfies f(x,y,z)= 3.Since coordinates of P satisfy the equation , it lies on the surface .
The tangent plane to the surface at P (2,1,3) is
is the tangent plane to the surface f(x,y,z)=3 at P(2,1,3).
For a parallel tangent plane, I understand that I need to find a point that satisfies the equation f(x,y,z)=3 AND an equation of a plane parallel to 6x-10y+z=5. However, I don't know yet how to find it.