Section 14.7: Max Min Problems

For functions of one variable it is impossible for a continuous function to have two local maxima and no local minimum. But for functions of two variables such functions exist. Show that the function \(f(x, y) = -(x^2 - 1)^2 - (x^2y - x - 1)^2\) has only two critical points, but has local maxima at both of them. Then use a computer to produce a graph with a carefully chosen domain and viewpoint to see how this is possible.