Determine the tangent planes of the level surface given by x^2+y^2+x^2 = 1 at the points (x,y,0) and show that they are all parallel to the z-axis.
here is my working thus far:
f(x,y,z) = x^2+y^2+z^2-1
then the tangent plane equation is:
(∂f/dx)|(x,y,0)(x-x) + (∂f/dy)|(x,y,0)(y-y) + (∂f/dz)|(x,y,0)(z-0) = 0
the first two terms go to zero since (x-x = y-y = 0), and the last term leaves 0z=0 (no information!)
Mathematically, where is my problem?