Download Lesson Nine Finding Extrema Given an Interval File

Extrema, Maxima
and Minima
Sketch the graph on the given interval:
f ( x)  x3  3x 2  9 x  11,  2  x  6
f x
80
Global Maximum
Value
70
60
50
40
Local Maximum
30
20
10
x
4
3
2
1
1
2
3
4
5
6
7
8
10
20
Global Minimum Value and
Local Minimum
Sketch the graph:
f ( x)  x3  3x 2  9 x  11
f x
80
70
60
50
40
Local Maximum
30
20
10
4
3
2
1
1
2
3
4
5
6
7
8
10
20
Local Minimum
x
f ( x)  x 2  1
Given the graph of
where f (x) is continuous on the
closed interval [-1, 2].
f x
6
Global, Local
Maximum Value
5
4
3
2
1
2
1
Global Minimum Value
and Local Minimum
1
1
2
2
3
x
Given the graph of
f ( x)  x 2  1
where f (x) is continuous on the
open interval ]-1, 2[.
f x
6
NO Maximum
Value
5
4
3
2
1
2
1
Global Minimum Value
and Local Minimum
1
1
2
2
3
x
Given the graph of
 x 2  1, x  0
where f (x) is
f ( x)  
2, x  0
on the closed interval [-1, 2].
Global, Local
Maximum Value
f x
6
5
4
3
2
1
NO Minimum Value
2
1
1
1
2
2
3
x
Given the graph of
f ( x)  x 2  1
where f (x) is continuous on the
set of all real numbers.
f x
6
NO Maximum
Value
5
4
3
2
1
2
1
Global Minimum Value
and Local Minimum
1
1
2
2
3
x
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.
Guidelines for finding maximums and minimums on a
closed interval:
1. Find the critical numbers of f in ]a, b[
2. Evaluate f at each critical number in ]a, b[
3. Evaluate f at each endpoint of [a, b]
4. The least of these values is the global minimum. The greatest is the global
maximum.
5. The greatest of any critical points of f in (a, b) is the local maximum.
The least of any critical points of f in (a, b) is the local minimum.
Example 1:
Find the maximums and minimums of the function below on
the interval [-1, 2] f ( x)  3x 4  4 x3
Local Maximum –
f x
20
(2, 16)
16
Local Minimum –
(1, -1)
12
Global Maximum – 16
8
4
Global Minimum -
-1
2
1
1
4
2
3
x
Example 2:
Find the extrema of the function below on the interval [-1, 2]
f ( x)  x  x
3
3
2
2
f x
3
Local Maximum –
(0, 0)
2
Local Minimum –
1
(1, -1)
2
Global Maximum – 2
1
1
1
2
Global Minimum -
-2.5
3
4
2
3
x
Example 3:
Find the extrema of the function below on the interval ]1, 3]
f ( x)  x  2 x
2
f x
4
Local Maximum –
(3, 3)
3
Local Minimum –
2
None
1
Global Maximum – 3
1
Global Minimum -
None
1
1
2
2
3
4
x