1. Relative Extrema Lesson 5.5

2. Video Profits Revisited Recall our Digitari manufacturer Cost and revenue functions C(x) = 4.8x -.0004x 2 0 ≤ x ≤ 2250 R(x) = 8.4x -.002x 2 0 ≤ x ≤ 2250 Cost, revenue, and profit functions 2

3. Video Profits Revisited Digitari wants to know how many to make and sell for maximum profit 3 Profits increasing on this interval Slope > 0 Profits increasing on this interval Slope > 0 Profits decreasing on this interval Slope < 0 Profits decreasing on this interval Slope < 0 Maximum profit when Profits neither increasing nor decreasing Slope = 0 Maximum profit when Profits neither increasing nor decreasing Slope = 0

4. Relative Maximum Given f(x) on open interval (a, b) with point c in the interval Then f(c) is the relative max if f(x) ≤ f(c) for all x in (a, b) 4 a b ( ) c

5. Relative Minimum Given f(x) on open interval (a, b) with point c in the interval Then f(c) is the relative min if f(x) ≥ f(c) for all x in (a, b) 5 a b ( ) c

6. Relative Max, Min Note Relative max or min does not guarantee f '(x) = 0 Important Rule: If a function has a relative extremum at c Then either c a critical number or c is an endpoint of the domain 6

7. First Derivative Test Given f(x) differentiable on (a, b), except possibly at c c is only critical number in interval f(c) is relative max if f '(x) > 0 on (a, c) and f '(x) < 0 on (c, b) 7 a b () c

8. First Derivative Test Given f(x) differentiable on (a, b), except possibly at c c is only critical number in interval f(c) is relative min if f '(x) < 0 on (a, c) and f '(x) > 0 on (c, b) 8 a b ( ) c

9. First Derivative Test Note two other possibilities f '(x) < 0 on both sides of critical point f '(x) > 0 on both sides of critical point Then no relative extrema 9

10. Finding Relative Extrema Strategy Find critical points Check f '(x) on either side Negative on left, positive on right → min Positive on left, negative on right → max Try it! 10

11. Application Back to Digitari … cost and revenue functions C(x) = 4.8x -.0004x 2 0 ≤ x ≤ 2250 R(x) = 8.4x -.002x 2 0 ≤ x ≤ 2250 Just what is that number of units to market for maximum profit? What is the maximum profit? 11

12. Assignment Lesson 5.2 Page 327 Exercises 1 – 53 EOO 12