5.2 Relative Extrema Find Relative Extrema of a function using the first derivative Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley

If a function f has a relative extreme value f (c) on an open interval; then c is a critical value. So, f (c) = 0 or f (c) does not exist. Relative extrema will only occur at points where the derivative is = 0 or where it is undefined. Relative Extrema

Relative Minimum Let I be the domain of f : f (c) is a relative minimum (bottom of a valley) if there exists within I an open interval I 1 containing c such that f (c) ≤ f (x) for all x in I 1 ; f has a relative minimum at c if f (x) < 0 on (a, c) and f (x) > 0 on (c, b). That is, f is decreasing to the left of c and increasing to the right of c.

If the graph is continuous (no break) at the point where the function changes from decreasing to increasing, that point is called a relative minimum point

Relative Maximum Let I be the domain of f : F (c) is a relative maximum (top of a hill) if there exists within I an open interval I 2 containing c such that f (c) ≥ f (x) for all x in I 2. f has a relative maximum at c if f (x) > 0 on (a, c) and f (x) < 0 on (c, b). That is, f is increasing to the left of c and decreasing to the right of c.

If the graph is continuous (no break) at the point where the function changes from increasing to decreasing, that point is called a relative maximum point.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Graph over the interval (a, b) f (c) Sign of f (x) for x in (a, c) Sign of f (x) for x in (c, b) Increasing or decreasing Relative minimum Relative maximum –+–+ +–+– Decreasing on (a, c]; increasing on [c, b) Increasing on (a, c]; decreasing on [c, b)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Graph over the interval (a, b) f (c) Sign of f (x) for x in (a, c) Sign of f (x) for x in (c, b) Increasing or decreasing No relative maxima or minima –+–+ –+–+ Decreasing on (a, b) Increasing on (a, b)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 1: For the function f given by find the relative extrema.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 1 (continued): Find Derivative And set it = 0 These two critical values partition the number line into 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Example 1 (continued): 3 rd analyze the sign of f (x) in each interval. Test Valuex = –2x = 0x = 4 Sign of f (x) +–+ Result f is increasing on (–∞, –1] f is decreasing on [–1, 2] f is increasing on [2, ∞) Using First Derivatives to Find Maximum and Minimum Values

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 1 (concluded): Therefore, by the First-Derivative Test, f has a relative maximum at x = –1 given by The relative maximum value is 19. It occurs where x = -1. And f has a relative minimum at x = 2 given by The relative minimum value is -8. It occurs where x is 2.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 1 (continued):

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 3: Find the relative extrema for the Function f (x) given by Then sketch the graph. 1 st find f (x).

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 3 (continued): 2 nd find where f (x) does not exist or where f (x) = 0. Note that f (x) does not exist where the denominator equals 0. Since the denominator equals 0 when x = 2, x = 2 is a critical value. f (x) = 0 where the numerator equals 0. Since 2 ≠ 0, f (x) = 0 has no solution. Thus, x = 2 is the only critical value.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 3 (continued): 3 rd x = 2 partitions the number line into 2 intervals: A (– ∞, 2) and B (2, ∞). So, analyze the signs of f (x) in both intervals. Test Valuex = 0x = 3 Sign of f (x) –+ Result f is decreasing on (– ∞, 2] f is increasing on [2, ∞)

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 3 (continued): Therefore, by the First-Derivative Test, f has a relative minimum at x = 2 given by The relative minimum value is 1. It occurs at x = 2.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide Using First Derivatives to Find Maximum and Minimum Values Example 3 (concluded): We use the information obtained to sketch the graph below, plotting other function values as needed.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide More Examples

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Slide C(q) above is the cost function. p(q) is the price function. Find a)The number of units that will produce a maximum profit. b)The maximum profit. c)The price that will produce a maximum profit. a) 1625 b) $ c) $57.50