Answer Happy

For functions of a single variable, there is a theorem that says if a differentiable function has two local maxima, then it must have a local minimum (this is easily proven using the first derivative test). However, this is not true in higher dimensions. (a) Show that the function f(x, y) = −(x² − 1)² – (x²y - x - 1)² has exactly two critical points, both of which are local maxima. (b) Use 3D graphing software to plot the graph of f(x, y) with a well- chosen domain and viewpoint to see how this is possible. Include a picture of this graph with your assignment. Note: The website www.math3d.org has a decent, although basic, 3D graphing calculator. To include a picture of the graph, you can just take a picture with your phone when you are scanning your homework, or you can take a screenshot and save is as an image, then use a website like www.pdfescape.com to insert the image into the .pdf of your homework (or just print it out or something, if you are old school).