1. Chapter 3 Applications of Differentiation Maximum Extreme Values
Minimum
Maximum
Extreme Values
(extrema)

2. 3.1 Extrema on an Interval Objectives
Geometrically (by Extreme Value Theorem) define absolute (global) extrema of a function on a closed interval.
Define the relative (local) extrema of a function on an open interval.
Compare and contrast relative and absolute extrema.
Define, identify, and find critical numbers and extrema through the use of derivatives.
Find relative and absolute extrema of a function using analytical and differentiation techniques

3. Defintion of Relative Extrema

4. The Extreme Value Theorem

5. Closed vs Open Intervals

6. Guarantee
The idea of a closed interval “guarantees” a minimum & maximum.

7. Relative Extrema & Critical Numbers
Extrema “relative” to a neighborhood of a function.

8. Definition of Relative Extrema

9. Finding Relative Extrema
f’(x) is undefined
Find “critical numbers” (values of x where
f’(x) is either zero or undefined.)

11. Relative extrema occur only at critical numbers.

12. The converse is not true
f (x) has a critical value at x = 2
because f ’ (2) = 0.
There is no extrema at x = 2.

13. Finding Extrema on the Closed Interval [-1,2]

14. 1.) Find critical numbers (-1,2)
1.) Find critical numbers by
identifying x-values for which
f ’(x) = 0 or is undefined.
f ’(x)

15. 2. Evaluate f at each critical number in (-1, 2)
x = 0
Critical number x = 1
f(0) = 0
f(1) = -1

16. 2. Evaluate f at each endpoint of [-1, 2]
x = -1
x = 2
f(-1) = 7
f(2) = 16

17. Finding Extrema on the Closed Interval [-1,2]
Endpoint
x = -1
f(-1) = 7
(2,16)
Maximum
(-1,7)
Critical number
x = 0
f(0) = 0
Critical number
x = 1
f(1) = -1
minimum
(0,0)
(1,-1)
Minimum
Endpoint
x = 2
f(2) = 16
maximum

18. Guidelines to finding extrema on a closed interval