Homework Set 16 Section 11.7: Maximum and Minimum Values 1. Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In each of the following cases, what can you say about f ? (a) fxx (1, 1) = 4, fxy (1, 1) = 1, and fyy (1, 1) = 2 (b) fxx (1, 1) = 4, fxy (1, 1) = 3, and fyy (1, 1) = 2 2. Find the local maximum and minimum value(s) as well as any saddle point(s) of the function f (x, y) = xy − 2x − 2y − x2 − y 2 . [Use a 3D plotting program to graph the function and verify your critical points. Be sure to set a viewing window that includes all of the critical points.] 3. Find the local maximum and minimum value(s) as well as any saddle point(s) of the function f (x, y) = xy + x1 + y1 . [Use a 3D plotting program to graph the function and verify your critical points.] Math 2241 hw set 16, page 2 of 2 due June 13 4. Find the absolute maximum and minimum values of f (x, y) = xy 2 on the domain D = {(x, y)|x ≥ 0, y ≥ 0, x2 + y 2 ≤ 3}. 5. 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 a situation is possible. Show that the function f (x, y) = −(x2 − 1)2 − (x2 y − x − 1)2 has only two critical points and has local maxima at both of them. Then use a 3D plotting program to graph the function (be careful on what viewing window you choose). What do you see happening in the graph that makes this possible? Is that something that could occur in 2D?

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