1. AP Calculus AB Chapter 3, Section 1
Extrema on an Interval

2. Extrema on an Interval
Absolute Extrema:
Relative or Local Extrema:

3. Extreme Value Theorem
If f is continuous on a closed interval [a, b], the f has both an absolute minimum and an absolute maximum on the interval.
The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. They are also called the absolute minimum or absolute maximum on the interval, respectively.

4. Find the absolute extrema of each function on the given interval.

5. Critical Numbers
A number x = c in the interior of the domain of a function f(x) is called a _______________ if either
𝑓 ′ 𝑐 = 𝑜𝑟 𝑓 ′ 𝑐 =𝐷𝑁𝐸

6. Definition of a Critical Number
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.

7. Critical Numbers
If f(x) is a smooth curve, there are three possible outcomes that can occur at a critical number:
An absolute or relative maximum can occur,
An absolute or relative minimum can occur,
A point of inflection can occur

8. Find any critical numbers of the function
𝑓 𝑥 = 2𝑥+5 3
𝑓 𝑥 = 𝑥 2 +2𝑥−4
𝑓 𝑥 = 𝑥−1 +5

9. Find the value of the derivative at each of the relative extrema for the function in the interval [-1, 3]. 𝑓 𝑥 = 𝑥 3 −3 𝑥 2

10. Find the value of the derivative at each of the relative extrema in the interval [0, 6] for the function 𝑓 𝑥 = 9( 𝑥 2 −3) 𝑥 3

11. Find the value of the derivative at each of the relative extrema for the function in the interval [-2, 2]. 𝑓 𝑥 = 𝑥

12. Find the value of the derivative at each of the relative extrema in the interval [0, 2𝜋] for the function 𝑓 𝑥 = sin 𝑥

13. Finding Extrema on a Closed Interval
Guidelines for finding extrema on a closed interval:
Find the critical numbers of f in (a, b).
Evaluate f at each critical number in (a, b).
Evaluate f at each endpoint of [a, b].
The least of these values is the minimum, the greatest is the maximum.

14. Find the extrema of 𝑓 𝑥 =3 𝑥 4 −4 𝑥 3 on the interval [-1, 2]
Find the extrema of 𝑓 𝑥 =3 𝑥 4 −4 𝑥 3 on the interval [-1, 2]. Hint: Differentiate, then find all x-values where f’(x)=0 and all x-values when f’(x) DNE.

15. Find the extrema of 𝑓 𝑥 =2𝑥−3 𝑥 2/3 on the interval [-1, 3]

16. Find the extrema of 𝑓 𝑥 =2 sin 𝑥 − cos 2𝑥 on the interval [0, 2𝜋]

17. Chapter 3.1 Homework
Pg. 169 – 171, #’s: 3, 7, 19, 27, 41, 61