For a function of a single variable f(x), if f is continuous on an interval I, has only one critical point in I, and that critical point is a local minimum, then it is the absolute (or global) minimum. However, for functions of more than one variable things are not so nice.
Let . This is differentiable for all (x,y). In looking for critical points, we solve
The only point solving both of these equations is (0,1), and the second derivative test shows that this point is a local minimum for f. However, f(0,-3)=-17<f(0,1)=-1, so that this local minimum is not an absolute minimum. Here's a picture:
(The way I constructed this example was by starting with , which has a local minimum at (1,1) and a saddle point at (0,0), and then sent the saddle point off to infinity by replacing x by .)
Henry Wente (of the University of Toledo) has given me a polynomial with the same property: , which he found in the Math Monthly, vol. 100, no. 3, March 1993. The article is ``Counting Critical Points of Real Polynomials in 2 Variables", by Durfee et. al., pp. 255-271.