1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Extrema OBJECTIVES  Find absolute extrema using Maximum- Minimum Principle 1.  Find absolute extrema using Maximum- Minimum Principle 2. 6.1

2. Slide 2.4 - 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: Suppose that f is a function with domain I. f (c) is an absolute minimum if f (c) ≤ f (x) for all x in I. f (c) is an absolute maximum if f (c) ≥ f (x) for all x in I.

3. Slide 2.4 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The absolute max of a function is the largest possible value of the function. It may or may not be the same as a relative max of the function. The absolute min of a function is the smallest possible value of the function. It may or may not be the same as a relative min.

4. Slide 2.4 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Extreme Value Theorem A continuous function f defined over a closed interval [a, b] must have an absolute maximum value and an absolute minimum value over [a, b].

5. Slide 2.4 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When referring to an absolute value of a function, it is customary to consider the function on some closed interval [a, b]. In these cases the absolute value will occur at an endpoint of the interval or else at a relative max or min in [a, b].

6. Slide 2.4 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Maximum-Minimum Principle 1 Suppose that f is a continuous function defined over a closed interval [a, b]. To find the absolute maximum and minimum values over [a, b]: a) First find f (x). b) Then determine all critical values in [a, b]. That is, find all c in [a, b] for which f (c) = 0 or f (c) does not exist.

7. Slide 2.4 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Maximum-Minimum Principle 1 (continued) c) List the values from step (b) and the endpoints of the interval: a, c 1, c 2, …., c n, b. d) Evaluate f (x) for each value in step (c): f (a), f (c 1 ), f (c 2 ), …., f (c n ), f (b). The largest of these is the absolute maximum of f over [a, b]. The smallest of these is the absolute minimum of f over [a, b].

8. Slide 2.4 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Every function that is continuous on a closed interval [a, b] will have an absolute max and an absolute min on the interval. To find the absolute extrema: (1) find all critcal numbers on [a,b] (2)evaluate f at each critical number and also f(a) and f(b) (3)the largest value is the ab max and the smallest value is the ab min

9. Slide 2.4 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Find the absolute maximum and minimum values of over the interval [–2, ]. a) b) Note that f (x) exists for all real numbers.

10. Slide 2.4 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (concluded): c) –2, –1, 1, d)

11. Slide 2.4 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (concluded): Thus, the absolute maximum value of f (x) on [–2, ] is 4, which occurs at x = 1. The absolute minimum value of f (x) on [–2, ] is 0, which occurs at x = 2 and x = 1.

12. Slide 2.4 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Maximum-Minimum Principle 2 Suppose that f is a function such that f (x) exists for every x in an interval I, and that there is exactly one (critical) value c in I, for which f (c) = 0. Then f (c) is the absolute maximum value over I if f (c) is the relative maximum. or f (c) is the absolute minimum value over I if f (c) is the relative minimum.

13. Slide 2.4 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: Find the absolute maximum and minimum values of When no interval is specified, we consider the entire domain of the function. In this case, the domain is the set of all real numbers. a)

14. Slide 2.4 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (continued): b) c) Since there is only one critical value, we can apply Maximum-Minimum Principle 2. f(2) is a relative maximum, so it is the absolute maximum

15. Slide 2.4 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (concluded): so f (x) has an absolute maximum value of 4 at x = 2. f (x) has no absolute minimum value.

16. Slide 2.4 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A Strategy for Finding Absolute Maximum and Minimum Values: To find absolute maximum and minimum values of a continuous function over an interval: a) Find f (x). b) Find the critical values. c) If the interval is closed use Maximum-Minimum Principle 1.

17. Slide 2.4 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A Strategy for Finding Absolute Maximum and Minimum Values (concluded): d) If the interval is not closed, such as (–∞, ∞), (0, ∞), or (a, b), and the function has only one critical value, use Maximum-Minimum Principle 2. In such a case, if the function has a maximum, it will have no minimum; and if it has a minimum, it was have no maximum. e) If the interval is not closed and there is more than one critical value, we won’t have these.

18. Slide 2.4 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Find the absolute maximum and minimum values of over the interval (0, ∞). a) b) Since f (x) exists for all values of x in (0, ∞), the only critical values are those for which f (x) = 0.

19. Slide 2.4 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): c) The interval (0, ∞) is not closed, and the only critical value is Thus, we can use Maximum- Minimum Principle 2.

20. Slide 2.4 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): Since a relative minimum occurs at f (x) has an absolute minimum at

21. Slide 2.4 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): Thus, the absolute minimum of f (x) is which occurs at f (x) has no absolute maximum value.