Use Lagrange multipliers to find the maximum and minumum values of the function subject to the given constraint

$\displaystyle f(x, y, z) = xyz$

$\displaystyle g(x, y, z) = x^2+2y^2+3z^2 = 6$

$\displaystyle \nabla f = [yz, xz, xy]$

$\displaystyle \nabla g = [2x, 4y, 6z]$

$\displaystyle yz - \lambda 2x = 0$

$\displaystyle xz - \lambda 4y = 0$

$\displaystyle xy - \lambda 6z = 0$

$\displaystyle x^2+2y^2+3z^2 = 6$

I've gotten the system of equations. (shown above) I'm just not sure how to go about tackling this one. Any words of wisdom from someone more clever would be appreciated