This document defines and discusses absolute and local extreme values of functions. It states that a continuous function on a closed interval will have both an absolute maximum and minimum value. Absolute extrema can occur at endpoints or interior points, while local extrema occur at interior points where the derivative is 0 or undefined. To find the absolute extrema on an interval, evaluate the function at the endpoints and critical points, and the greatest and least values will be the absolute max and min.

1. 3.1
Extreme
Values of
Functions

2. Def: Absolute Extreme Values
Let f be a function with domain D. Then f(c)
is the…
a) Absolute (or global) maximum value on
D iff f(x) ≤ f(c) for all x in D.
b) Absolute (or global) minimum value on
D iff f(x) ≥ f(c) for all x in D.
* Note: The max or min VALUE refers to
the y value.

3. Where can absolute extrema
occur?
* Extrema occur at endpoints or interior points in an interval.
* A function does not have to have a max or min on an interval.
* Continuity affects the existence of extrema.
Min
Max
Min
No Max No Max
No Min

4. Thm 1: 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.
* What is the “worst case scenario,”
and what happens then?

5. Def: Local Extreme Values
Let c be an interior point of the domain of f.
Then f(c) is a…
a) Local (or relative) maximum value at c iff
f(x) ≤ f(c) for all x in some open interval
containing c.
b) Local (or relative) minimum value at c iff
f(x) ≥ f(c) for all x in some open interval
containing c.
* A local extreme value is a max or min in a
“neighborhood” of points surrounding c

6. Identify Absolute and Local
Extrema
Abs & Loc Min
Abs & Loc Max
Loc Max
Loc Min
Loc Min
* Local extremes are where f ’(x)=0 or f ’(x) DNE

7. Thm 2: 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 f ’ exists at
c, then f ’(c)=0.

8. Def: Critical Point
A critical point is a point in the interior
of the domain of f at which either…
1) f ’(x)= 0, or
2) f ’(x) does not exist

9. Finding Absolute
Extrema on [a,b]…
To find the absolute extrema of a
continuous function f on [a,b]…
1. Evaluate f at the endpoints a and b.
2. Find the critical numbers of f on [a,b].
3. Evaluate f at each critical number.
4. Compare values. The greatest is the
absolute max and the least is the
absolute min.

10. Check for Understanding…
(True or False??)
1. Absolute extrema can occur at either interior
points or endpoints.
2. A function may fail to have a max or min
value.
3. A continuous function on a closed interval
must have an absolute max or min.
4. An absolute extremum is also a local
extremum.
5. Not every critical point or end point signals
the presence of an extreme value.