1. EXTREMA ON AN INTERVAL
Section 3.1

2. When you are done with your homework, you should be able to…
Understand the definition of extrema of a function on an interval
Understand the definition of relative extrema of a function on an open interval
Find extrema on a closed interval

3. EXTREMA OF A FUNCTION

4. DEFINITION OF EXTREMA
Let f be defined on an open interval I containing c.
is the minimum of f on I if
for all x in I.
is the maximum of f on I if

5. EXTREMA CONTINUED…
The minimum and maximum of a function on an interval are the extreme values, or extrema of the function on the interval
The singular form of extrema is extremum
The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval

6. EXTREMA CONTINUED…
A function does not need to have a maximum or minimum (see graph)
Extrema that occur at endpoints of an interval are called endpoint extrema

7. THE EXTREME VALUE THEOREM
If f is continuous on a closed interval
then f has both a minimum
and a maximum on the interval.

8. DEFINITION OF RELATIVE EXTREMA
If there is an open interval containing c on which is a maximum, then is called a relative maximum of f, or you can say that f has a relative maximum at
If there is an open interval containing c on which f is a minimum, then is called a relative minimum of f, or you can say that f has a relative minimum at

9. Find the value of the derivative (if it exists) at the indicated extremum.
0.0

10. Find the value of the derivative (if it exists) at the indicated extremum.
0.0

11. Find the value of the derivative (if it exists) at the indicated extremum.

13. DEFINITION OF A CRITICAL NUMBER
Let f be defined at c.
If , then c is a critical number of f.
If f is not differentiable at c, then c is a critical number of f.

14. Locate the critical numbers of the function.
None of these

15. Locate the critical numbers of the function.
None of these

16. THEOREM: RELATIVE EXTREMA OCCUR ONLY AT CRITICAL NUMBERS
If f has a relative maximum or minimum at
, then c is a critical number of f.

17. GUIDELINES FOR FINDING EXTREMA ON A CLOSED INTERVAL
To find the extrema of a continuous function f on a closed interval , use the following steps.
Find the critical numbers of f in c.
Evaluate f at each critical number in
Evaluate f at each endpoint of
The least of these outputs is the minimum. The greatest is the maximum.

18. The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.
True
False