About the method of Lagrange multipliers to extremize a function

Question

Suppose that by the using the method of Lagrange multipliers to extremize the function $f(x,y)$ subject to the constraint $g(x,y)$, we find that $f(x,y)$ has only one critical point. How can we deduce the critical point is minumum or maximum?

Answer 1

Well, if your critical point is $(x_0,y_0)$ you can just look at any other point $(x,y)$ fulfilling the constraint $g(x,y)$.

If $f(x_0,y_0) > f(x,y)$ the critical point is a maximum, otherwise it is a minimum.