1. Extreme values of functions
Section 4.1a

2. Definition: Absolute Extreme Values
Let be a function with domain D. Then is the
absolute maximum value on D if and only if
for all x in D.
(b) absolute minimum value on D if and only if
for all x in D.
Absolute (or global) maximum and minimum values are also
called absolute extrema. Sometimes the word “absolute”
or “global” is simply omitted…
The Extreme Value Theorem (EVT)
If is continuous on a closed interval [a, b], then
has both a maximum value and a minimum value
on the interval.

3. How about a graph for each of these???
For our first example, let’s complete the following table:
Function
Domain D
Absolute Extrema on D
No absolute max.
Absolute min. of 0 at x = 0
Abs. max. of 4 at x = 2
Abs. min at (0, 0)
Abs. max. of 4 at x = 2
No absolute min.
No abs. extrema
How about a graph for each of these???

4. Definition: Local (Relative) Extreme Values
Let c be an interior point of the domain of the function .
Then is a
local maximum value at c if and only if
for all x in some open interval containing c.
(b) local minimum value at c if and only if
for all x in some open interval containing c.
Note: all absolute extrema are also local extrema!!!

5. x Abs. Max. Local Max. Local Min. Abs. Min. Local Min.
Let’s classify extreme values in the function shown above

6. Guided Practice – Identify Any Extreme Values7 3 x –7 –4 –2 5 10
Local Min of 0 at x = – 7 –6
Max of 7 at x = – 4 and x = 5
Min of – 6 at x = – 2
Local Min of 3 at x = 10

7. Theorem: Local Extreme Values
The “Do Now”
Writing: How can the derivative help in identifying
extreme values of functions (maxima and minima)?
Theorem: Local Extreme Values
If a function has a local maximum value or a local
minimum value at an interior point c of its domain,
and if exists at c, then

8. Definition: Critical Point
A point in the interior of the domain of a function
at which or does not exist is a
critical point of .
To find extrema analytically, we only
need to investigate the critical points
and the endpoints of the function!!!

9. Guided Practice No zeros, but undefined at x = 0
Find the extreme values of on the interval [–2, 3].
First, find the first derivative:
Next, identify any critical points:
No zeros, but undefined at x = 0
(i.e., the function has a critical point at x = 0)

10. Guided Practice  Abs. Min.  Local Max.  Abs. Max.
Find the extreme values of on the interval [–2, 3].
Finally, evaluate and analyze the original function at this
critical point, and at the endpoints:
Crit. Pt:
 Abs. Min.
Endpts:
 Local Max.
 Abs. Max.
Can we support these answers graphically???

11. Guided Practice  Find answer both analytically and graphically!
Find the extreme values of
 Find answer both analytically and graphically!
Domain is all real numbers…need to check critical points…
The derivative is never undefined.
The derivative is zero for x = 1.
As x moves away from 1 in either direction, f increases.
Absolute Minimum of 3 at x = 1

12. Maximum value is e at x = –1 Minimum value is 1/e at x = 1
Guided Practice
Find the extreme values of the given function.
The derivative has no zeros.
We only need to check endpoints.
Maximum value is e at x = –1
Minimum value is 1/e at x = 1