Method of Lagrange multipliers determining nature?

Question

Let us say I have a function, $g$ that I want to find the extrema of subject to the condition $h=0$ via the method of Lagrange multipliers. i.e. so that I find the extrema of the function: $$f=g-\lambda h$$ If $f$ is a maximum (say), can we say anything about the nature of the stationary point of $g$ (e.g. can we say that it is also a maximum, or that it can't be a minimum)? and similarly for $f$ been a minimum or stationary point?

Answer 1

I'm not fully sure I understand the question. I see a few interpretations:

• Are stationary points of $f$ stationary points of $g$? The answer is very much no: for instance, $g(x,y)=xy$ only has a stationary point at $(0,0)$, but that's not the only stationary point we find when we maximize $g$ subject to $2x+2y=10$. However, stationary points of $g$ which satisfy the constraint are certainly stationary points of $f$.

• When I find a maximum (resp. minimum) of $f$, do I have a maximum (resp. minimum) of $g$ under the constraint $h=0$? The answer is yes, but the converse is not true. It turns out that typically the stationary points in Lagrange multipliers are saddle points of $f$, even when we find extrema of $g$ under the constraint.

Answer 2

In general, no. The method of Lagrange multipliers finds points on the level surface $h=\text{const}$ at which the gradients of $g$ and $h$ are parallel, i.e., where $\nabla f$ is normal to this surface, but that need not happen at a stationary point of the unrestricted function $g$. Indeed, $g$ may not have any stationary points at all. Consider, for example, $g:(x,y)↦x+y$ with $h$ the unit circle.