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.

Mon May 5 12:53:33 CDT 1997