"Calculus and its applications: " - Calculus and its applications-Maximum-minimum problems have objective: find relative extrema of a function of two variables. | 2012 Pearson Education, Inc. All rights reserved Slide Maximum-Minimum Problems OBJECTIVE Find relative extrema of a function of two variables. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: A function f of two variables: 1. has a relative maximum at (a, b) if f (x, y) ≤ f (a, b) for all points in a rectangular region containing (a, b). 2. has a relative minimum at (a, b) if f (x, y) ≥ f (a, b) for all points in a rectangular region containing (a, b). Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1: The D-Test To find the relative maximum and minimum values of f: 1. Find fx, fy, fxx, fyy, and fxy. 2. Solve the system of equations fx = 0, fy = 0. Let (a, b) represent a solution. 3. Evaluate D, where D = fxx(a, b)·fyy(a, b) – [ fxy(a, b)]2. Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1 (concluded): 4. Then: a) f has a maximum at (a, b) if D > 0 and fxx(a, b) 0 and fxx(a, b) > 0. c) f has neither a maximum nor a minimum at (a, b) if D p. 453, formula 12, there is no “+C” 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Find the relative maximum or minimum values of 1. Find fx, fy, fxx, fyy, and fxy. Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (continued): 2. Solve the system of equations fx = 0 and fy = 0. Using substitution, Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (continued): Then, substituting back, Thus, (2, –1) is the only critical point. 3. Find D. Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (concluded): 4. Since D = 3 and fxx(2, –1) = 2, since D > 0 | 2012 Pearson Education, Inc. All rights reserved Slide Maximum-Minimum Problems OBJECTIVE Find relative extrema of a function of two variables. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITION: A function f of two variables: 1. has a relative maximum at (a, b) if f (x, y) ≤ f (a, b) for all points in a rectangular region containing (a, b). 2. has a relative minimum at (a, b) if f (x, y) ≥ f (a, b) for all points in a rectangular region containing (a, b). Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1: The D-Test To find the relative maximum and minimum values of f: 1. Find fx, fy, fxx, fyy, and fxy. 2. Solve the system of equations fx = 0, fy = 0. Let (a, b) represent a solution. 3. Evaluate D, where D = fxx(a, b)·fyy(a, b) – [ fxy(a, b)]2. Maximum-Minimum Problems 2012 Pearson Education, Inc. All rights reserved Slide THEOREM 1 (concluded): 4. Then: a) f has a maximum at (a, b) if
