Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x, y, z) = x - y +z$ on the sphere $x^2 + y^2 +z^2$.
Since gradient vectors have have $x$ to the power of 1 or less does this mean there is only one point?
So far I have this: $$ 1 = \lambda (2x) \\ -1 = \lambda (2y) \\ 1 = \lambda (2z) \\ x^2 y^2 +z^2 -1 =0 $$
Hence I only got the point $(\dfrac{1}{\sqrt3}, \dfrac{-1}{\sqrt3} , \dfrac{1}{\sqrt3} )$.