1. AP CALCULUS AB Chapter 4: Applications of Derivatives Section 4.1:
Extreme Values of Functions

2. What you’ll learn about
Absolute (Global) Extreme Values
Local (Relative) Extreme Values
Finding Extreme Values
…and why
Finding maximum and minimum values of a
function, called optimization, is an important
issue in real-world problems.

3. Absolute Extreme Values
Absolute maximum and minimum values are also called
absolute extrema. The term “absolute” is often omitted.

4. The Extreme Value Theorem

5. The Extreme Value Theorem
If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.
2 conditions for f: continuous & closed interval
If either condition does not exist the E.V.T. does not apply.

6. Classifying Extreme Values

7. Local Extreme Values

8. Local Extreme Values

9. Extreme Values can be Absolute or Local
Local (relative) Extreme Values
Let c be an interior point of the domain of the function f.
Then f(c) is a
Local maximum value at c if and only if
f(x) ≤ f(c) for all x in some open interval containing c.
b) Local minimum value at c if and only if
f(x) ≥ f(c) for all x in some open interval containing c.
A function f may have a local max or local min at an endpoint c if the appropriate inequality holds.
An extreme value may be local and global.

10. Critical Points

11. Section 4.1 – Extreme Values of Functions
Not all critical numbers (points) may be actual relative minimums or maximums.
f’(c)=0, but
no max or min
minimum where
f’(c) does not exist

12. Example Finding Absolute Extrema

13. Example Finding Extreme Values

14. Section 4.1 – Extreme Values of Functions
Theorem: Local Extreme Values
If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if exists at c, then
In other words, if f has a relative maximum or relative minimum at x = c, then c is a critical number of f.

15. Section 4.1 – Extreme Values of Functions
(Absolute)
Finding Extrema on a Closed Interval
Find the critical numbers of f in
Evaluate f at each critical number in
Evaluate f at each endpoint of
The least of these values is the absolute minimum.
The greatest of these values is the absolute maximum.

16. Graphically Look at the graph.
The highest point on the graph is the global / absolute maximum.
The lowest point on the graph is the global / absolute minimum.
Don’t forget to consider the endpoints when looking at a closed interval!
Find the maximum and minimum points of f(x) = cos x on [-π, π]

17. Example 3 Finding Absolute Extrema
Find the absolute maximum and minimum values of f(x) = x 2/3 on the interval [-2,3].
Find x values where f ’ = 0 (make sure they are in the domain)
Check endpoints of domain
Find the y-values of each critical point to determine the maximum and minimum points.
Practice: Find absolute extrema of g(x)=e-x on [-1,1].

18. Finding Extreme Values
Find the extreme values of
Derivative = 0
Endpoints
Find y-values to evaluate

19. Make Sure to Answer the Question being Asked!
What is the maximum value? 
The output value is crucial.
Where does the maximum occur? 
The input value is crucial.
Where on the curve is the maximum? 
The POINT (ordered pair) is crucial.

20. Summary
Absolute extrema is on a closed interval and can be at the critical numbers (derivative = 0 or undefined) or at the endpoints of the interval.
Relative/local extrema are the maximum/minimum values that the function takes on over smaller open intervals. There are no endpoints to test like we do for absolute extrema.

21. You try:
What are the extreme values of

22. You try:
Find the absolute maximum and minimum values of

23. Section 4.1 – Extreme Values of Functions
Finding the Relative (local) maximum or minimum of a function (using a graphing calculator)
Enter the equation in y=.
Graph in the appropriate window.
Use (2nd) CALC, maximum or minimum (On the TI-89, use F5 (Math), max or min).
Arrow to just left of the critical point, ENTER.
Arrow to just right of the critical point, ENTER.
Guess? – Just press ENTER

24. Section 4.1 – Extreme Values of Functions
To find approximations of the roots of a polynomial using a graphing calculator:
Enter the equation in y=.
Graph in the appropriate window.
Use (2nd) CALC, ZERO (on the TI-89, use F% (Math) ZERO).
Arrow to just left of the zero, ENTER.
Arrow to just right of the zero, ENTER.
GUESS? – press ENTER.