Chapter 3
Applications of
Differentiation f ( x )

3 x
3 
6 x
2 
3
• Minimum
• Maximum
Extreme Values
(extrema)

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

Defintion of Relative Extrema

The Extreme Value Theorem

Closed vs Open Intervals

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

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

Definition of Relative Extrema

Finding Relative Extrema f’(x) is undefined f ' ( x )

0 f ' ( x )

0
Find “critical numbers” (values of x where
f’(x) is either zero or undefined.)

Relative extrema occur only at critical numbers.

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.

Finding Extrema on the Closed Interval [-1,2] f ( x )

3 x
4 
4 x
3

1.) Find critical numbers (-1,2) f ( x )

3 x
4 
4 x
3
1.) Find critical numbers by identifying x-values for which f ’(x) = 0 or is undefined.
f ’(x)

12 x
3 
12 x
2 
0
12 x
2
( x

1 )

0 x

0 , 1

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

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

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

Guidelines to finding extrema on a closed interval