Suppose that $x,y,z \in \mathbb{R}_{\ge 0}$. Consider $f(x,y,z) = xyz$ (the volume of some cube) and $g(x,y,z) = x+y+z-1$. I wish to maximise $f$ subject to $g=0$ with Lagrange multipliers.
The zero locus of $g$ is clearly compact, so max/min of $f$ exists. As $\nabla f = (yz,xz,xy), \nabla g = (1,1,1)$, there should be $\lambda \in \mathbb{R}$ s.t. $(yz,xz,xy) = \lambda(1,1,1)$. This implies either:
- two of $x,y,z$ are $0$ (minimum of $f$), or
- $x=y=z$ (maximum of $f$).
My problem is, $z=0$ is also a minimum, but Lagrange multipliers is not catching that! Did I miss anything? Thank you!