From: cleary@zimmer.csufresno.edu (Sean Cleary) Newsgroups: sci.math.research Subject: Re: Crititical Points of Polynomials Date: 17 Mar 1998 21:35:35 GMT The example that Teck Cheong Lim remembers from Stewart's calculus book is f(x,y)= -(x^2-1)^2-(x^2y-x-1)^2 which has the property that there are only two critical points and both are local maxima. There are no saddles or local minima. I used Mathematica to plot it from a viewpoint where you can see the shape and put it on my web page as http://zimmer.csufresno.edu/~cleary/humps.html Also see the problem "Two Mountains without a Valley", proposed and solved by Ira Rosenholz, Mathematics Magazine, Vol 60, No 1, Feb 1987 p.48, referenced in Anton's Calculus book, which gives an analytic solution. Sean Cleary Department of Mathematics Peters Building 359 CSU Fresno CA 93740 sean_cleary@csufresno.edu http://zimmer.csufresno.edu/~cleary ============================================================================== [The polynomial y^5 + x^2y^3 - y has a local maximum, a local minimum, and no other critical points. For graph, analysis, and background, see http://www.mtholyoke.edu/~adurfee/reu/89/reu89.htm --djr]