1. Extreme Values of Functions
4.1
Extreme Values of Functions

2. Quick Review

3. Quick Review

4. Quick Review Solutions

5. Quick Review Solutions

6. What you’ll learn about
Absolute (Global) Extreme Values
Local (Relative) Extreme Values
Finding Extreme Values
Essential Questions
How do we find the maximum and minimum values of a function, called optimization?
How do we use this in real-world problems?

7. Absolute Extreme Values
1. Find the absolute extrema for the following functions on the given domain.
Function Rule
Domain D
Absolute Extrema on D
No absolute maximum.
Absolute minimum of 0 at x = 0.
Absolute maximum of 4 at x = 2.
Absolute minimum of 0 at x = 0.
Absolute maximum of 4 at x = 2.
No absolute minimum.
No absolute extrema.

8. The Extreme Value Theorem

9. Classifying Extreme Values

10. Local Extreme Values

11. Local Extreme Values Critical Points
2. Find the absolute maximum and minimum values of
on the interval [-2, 3].
Critical value at x = 0.
Absolute Maximum is 2.08 at x = 3.
Absolute Minimum is 0 at x = 0.

12. 3.
Since the graph has no endpoints, all the extreme values must occur at critical points.
Domain: (-3, 3)
The only critical value is at x = 0.
Absolute Minimum is 1/3 at x = 0.
No Maxima, either local or absolute.

13. Pg. 193, 4.1 #1-41 odd