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?

Perhaps something like this: 
$$ yz = 2x \lambda |\times  x\\
xz = 2y \lambda |\times y \\
xy = 2z \lambda |\times z\\ $$
$$ xyz = 2x^2 \lambda\\
xyz = 2y^2 \lambda \\
xyz = 2z^2 \lambda \\ $$
Now we can something like this for example divide firs two equations and get: 
$$ x^2=y^2=z^2 $$
Now solve this step by step - consider octants for each case and you'll get a bunch of solutions I guess. Symmetrical equations always include something bit more than the ordinary ones. 
