Difficult time finding critical points using Lagrange The function is $f(x,y,z) = xyz$ on $x^2 + y^2 + z^2 = 1$. So I have:
$yz = 2x \lambda \\
xz = 2y \lambda \\
xy = 2z \lambda \\
x^2 + y^2 + z^2 = 1$
I guessed $x = \pm 1, y = 0, z = 0, \lambda = 0$ but apparently this isn't a critical point. There's many more that I can't seem to find using algebra either. I can't divide anything out because I can't divide by $0$, and I also can't factor anything. How can I find all the critical points?

The definition of critical points are all points satisfying $\nabla f=0$, which is 
$\partial f /\partial x = 0$
$\partial f /\partial y = 0$
$\partial f /\partial z = 0$
Therefore, the critical points you want to find should satisfy the following nonlinear system:
$yz=0$
$xz=0$
$xy=0$
$x^2+y^2+z^2=1$
Clearly, there exist lots of critical points. There is no closed form solutions for this system. 
