Use Lagrange multipliers to find the maximum and the minimum of f subject to the given constraint(s) $f(x,y)=xyz$ such that $x^2+y^2+z^2=3$.
So far we have $$\begin{align} f(x,y)&=xyz\\ g(x,y)&=x^2+y^2+z^2-3\\ f_x&=yz\\ f_y&=xz\\ \lambda g_x&=2x\lambda\\ \lambda g_y&=2y\lambda\\ f(x,y)&=2x^2\lambda\\ f(x,y)&=2y^2\lambda, \end{align}$$
so $x=y$. So I figured $x$ has to equal $1$ or $-1$. So the max is $1$ and the min is $-1$, but is this the right way to go about the problem?