1. Section 3.1 Day 1 Extrema on an Interval
AP Calculus AB

2. Learning Targets
Define the terms absolute extrema, relative extrema, absolute maximum/minimum, relative maximum/minimum, critical number
Determine the absolute extrema and relative extrema from a graph
Determine the absolute extrema analytically of a function
Define and apply the Extreme Value Theorem
Determine the critical numbers of a function
Communicate the relationship between the relative extrema and critical numbers

3. The maximum function value on the given interval Extrema
Absolute Maximum
The maximum function value on the given interval
Extrema
The minimum & maximum values of the function on the given interval
Absolute Minimum
The minimum function value on the given interval

4. The maximum function value on the open interval Relative Extrema
Relative Maximum
The maximum function value on the open interval
Relative Extrema
The minimum & maximum values of the function on an open interval
Relative Minimum
The minimum function value on the open interval

5. Key Point
It is possible for a function to not have a maximum or a minimum. For example…
Absolute Extrema: the function could have a hole at those points
Relative Extrema: the function could be continuously increasing or decreasing over the entire real number line

6. Extreme Value Theorem
If 𝑓 is continuous on a closed interval [𝑎,𝑏], then 𝑓 has both a minimum and a maximum on the interval

7. Relative Extrema
In your groups, use the knowledge you have learned from calculus this year to determine what is the mathematical similarity between the relative extrema on this graph.

8. Critical Numbers
Let 𝑓 be defined at 𝑐. If 𝑓 ′ 𝑐 =0 or if 𝑓 is not differentiable at 𝑐, then 𝑐 is a critical number of 𝑓.

9. Relative Extrema & Critical Numbers
If 𝑓 has a relative extrema at 𝑥=𝑐, then 𝑐 is a critical number of 𝑓.
In other words, relative extrema only occur at critical numbers.

10. Example 1
Determine if the graph will consist of both absolute and relative extrema or just relative extrema. Support your stance.
Determine the coordinate points of the extrema of the function

11. Example 2
Determine if the graph will consist of both absolute and relative extrema or just relative extrema. Support your stance.
Determine the coordinate points of the extrema of the function

12. Procedure for finding Absolute Extrema on a Closed Interval
Find the critical numbers of 𝑓 in the interval
Evaluate 𝑓 at each critical number in the interval
Evaluate 𝑓 at the endpoints of the interval
Determine where the absolute extrema occur at

13. Example 3
Find the absolute extrema of 𝑓 𝑥 =3 𝑥 2 −4 𝑥 3 on the interval [−1, 2] 1. 𝑓 ′ 𝑥 =6𝑥−12 𝑥 2 =6𝑥(1−2𝑥) 2. 𝑓 ′ 𝑥 =0 ⇒ 𝑥=0, 𝑥= −1, 7 , 0, 0 , 1 2 , 1 4 , 2, −20 Min (2, −20) Max (−1, 7)

14. Example 4
Find the absolute extrema of 𝑓 𝑥 =2𝑥−3 𝑥 on the interval [−1, 3]
𝑓 ′ 𝑥 =𝑥−2 𝑥 − 1 3
𝑓 ′ 𝑥 =0⇒𝑥= 4 8
𝑓 ′ 𝑥 =𝐷𝑁𝐸⇒𝑥=0
Min (−1, −5)
Max (0, 0)

15. Example 5
Find the absolute extrema of 𝑓 𝑥 = 𝑡 2 𝑡 2 +3 on the interval [−1, 1] Min (0, 0) Max −1, 1 4 , 1, 1 4

16. Exit Ticket for Feedback
Find the absolute extrema of 𝑓 𝑥 =− 𝑥 2 +3𝑥 on the interval [0, 3]