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Maximum-Minimum Problems
6.3
OBJECTIVE
• Find relative extrema of a function of two variables.
2012 Pearson Education, Inc.
All rights reserved Slide 6.3-2
6.3Maximum-MinimumProblems
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).
2012 Pearson Education, Inc. All rights reserved Slide 6.3-3
6.3Maximum-MinimumProblems
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.
2012 Pearson Education, Inc. All rights reserved Slide 6.3-4
6.3Maximum-MinimumProblems
THEOREM 1 (concluded):
4. Then:
a) f has a maximum at (a, b) if D > 0 and fxx(a, b) < 0. b) f has a minimum at (a, b) if D > 0 and fxx(a, b) > 0. c) f has neither a maximum nor a minimum at (a, b) if
D < 0. The function has a saddle point at (a, b). d) This test is not applicable if D = 0.
2012 Pearson Education, Inc. All rights reserved Slide 6.3-5
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