1. Relative Extrema
Lesson 5.2

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

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

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) c b ( ) a

6. Relative Max, Min Note Important Rule:
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

7. First Derivative Test Given f(c) is relative max if
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) ( ) a c b

8. First Derivative Test Given f(c) is relative min if
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) c b ( ) a

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

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!

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

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