4.1 Extreme Values of
Functions
Absolute extreme Values
• Absolute extreme values are maximum or
minimum y-values of a function
• These are also sometimes called global
values
• Does every function have absolute
extreme values? Give examples
Extreme values can be in the interior or the end points of
a function.
4
No Absolute
Maximum
3
2
1
yx
2
D   ,  
-2
-1
0
1
2
Absolute Minimum

4
Absolute
Maximum
3
2
1
yx
2
D  0, 2
-2
-1
0
1
2
Absolute Minimum

4
Absolute
Maximum
3
2
1
yx
2
D   0, 2
-2
-1
0
1
2
No Minimum

4
No
Maximum
3
2
1
yx
2
D   0, 2 
-2
-1
0
1
2
No Minimum

Local Extreme Values:
A local maximum is the maximum value within some
open interval.
A local minimum is the minimum value within some open
interval.

Absolute maximum
(also local maximum)
Local maximum
Local minimum
Local minimum
Absolute minimum
(also local minimum)
Local extremes
are also called
relative extremes.

Absolute maximum
(also local maximum)
Local maximum
Local minimum
Notice that local extremes in the interior of the function
occur where f  is zero or f  is undefined.

Extreme Value Theorem:
If f is continuous over a closed interval [a,b], then f
has both an absolute maximum and minimum value
over that interval.
Maximum &
minimum
at interior points
Maximum &
minimum
at endpoints
Maximum at
interior point,
minimum at
endpoint

Notice that extreme values occur where the
derivative is zero or undefined or at the endpoints
f '0
f ' DNE
f '0
Maximum &
minimum
at interior points
Maximum &
minimum
at endpoints
Maximum at
interior point,
minimum at
endpoint

Critical Point:
A point in the domain of a function f at which f   0
or f  does not exist is a critical point of f .
Note:
Maximum and minimum points in the interior of a function
always occur at critical points, but critical points are not
always maximum or minimum values.

Critical points are not always extremes!
2
yx
3
1
-2
-1
0
1
2
f0
-1
(not an extreme)
-2

2
yx
1/ 3
1
-2
-1
0
1
2
f  is undefined.
-1
(not an extreme)
-2
p
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
i. e. c is a critical number of f

To find absolute extreme values of a
continuous function over a closed interval
[a,b]
1. Take the derivative of f
2. Find the critical points of f
3. Evaluate the function at the critical
points and endpoints
4. The highest value is the absolute
maximum and the lowest value is the
absolute minimum
Examples
• Find the critical numbers
f  x   x 2  x  3
y  x 4x
y
4x
x2  1
Examples
• Find the absolute extreme values over the
interval
f  x   x 2  2x  4; [2, 1]
f  x   x 2  3x; [0, 3]
2
3
f  x   3 x  2 x; [ 1, 1]
1
f  x   cos p x; [0, ]
6
1
Finding Maximums and Minimums
of any function over its domain:
Find the derivative of the function, and determine
where the derivative is zero or undefined. These
are the critical points.
2
Find the value of the function at each critical point.
3
Find values or slopes for points between the
critical points to determine if the critical points are
maximums or minimums.
4
For closed intervals, check the end points as
well.

Examples: Find the extreme values
f  x   x 3  2x  4
f  x   x 3  3x  3x  2
1
f x  2
x 1
x 1
f x  2
x  2x  2