I was trying to find min/max for $f(x,y,z)=x^2y^2z^2$ subject to $x^2+y^2+z^2=1$. I´m working with Stewart's Calculus, and I haven´t seen yet second order conditions.
I found out that the function has 9 critical points at $\left(\frac{1}{3}^{1/2},\frac{1}{3}^{1/2},\frac{1}{3}^{1/2}\right)$ (the values of $x$, $y$ and $z$ could be positive or negative). Those are clearly max.
Then I checked on this website that has every Stewart solution, and there the point $(0,0,0)$ is shown as a minimum. Is that correct? Why? It doesn't satisfy $x^2+y^2+z^2=1$.