3.1 Extrema On An Interval
After this lesson, 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
Definition
Extrema
Minimum and maximum values on an interval are called
extremes, or extrema on an interval.
• The minimum value of the function on an interval is
considered the absolute minimum on the interval.
• The maximum value of the function on an interval is
considered the absolute maximum on the interval.
When the just word minimum or
maximum is used, we assume it’s an
absolute min or absolute max.
OPEN intervals – Do the following have extrema?
On an open
interval, the max.
or the min. may or
may not exist even
if the function is
continuous on this
interval.
CLOSED intervals – Do the following have extrema?
On a closed
interval, both
max. and min.
exist if the
function is
continuous on this
interval.
The Extreme Value Theorem (EVT)
Theorem 3.1: If f is continuous on a closed interval [a, b],
then f has both a minimum and a maximum on the interval.
In other words, if f is continuous on a closed interval, f
must have a min and a max value.
Max-Min
f is continuous
on [a, b]
a
b
Example
Example 1 Let f (x) = x2 – 5x – 6 on the closed interval
[–1, 6], find the extreme values.
Example
Example 2 Let f (x) = x3 + 2x2 – x – 2 on the closed
interval [–3, 1], find the extreme values.
Example
Example 3 Let f (x) = x3 + 2x2 – x – 2 on the closed
interval [–3, 2], find the extreme values.
The (absolute)max and (absolute)min of f on [a, b]
occur either at an endpoint of [a, b] or at a point in (a,
b).
Relative Extrema and Critical Numbers
(AP may use Local Extrema)
1. If there is an open interval containing c on which f (c) is a
maximum, then f (c) is a local maximum of f.
2. If there is an open interval containing c on which f (c) is a
minimum, then f (c) is a local minimum of f.
When you look at the entire graph (domain),
there may be no absolute extrema, but there
could be many relative extrema.
What is the slope at each extreme value????
Definition of a Critical Number and Figure 3.4
Critical Numbers
c is a critical number for f iff:
1.
f (c) is defined (c is in the domain of f )
2.
f ’(c) = 0 or f ’(c) = does not exist
Theorem 3.2 If f has a relative max. or relative min,
at x = c, then c must be a critical number for f.
The (absolute)max and (absolute)min of f on [a, b]
occur either at an endpoint of [a, b] or at a critical
number in (a, b).
So…. Relative extrema can only occur at critical values, but
not all critical values are extrema. Explain this statement.
Guidelines
Make sure f is continuous on [a, b].
1. Find the critical numbers of f(x) in (a, b). This is
where the derivative = 0 or is undefined.
2. Evaluate f(x) at each critical numbers in (a, b).
3. Evaluate f(x) at each endpoint in [a, b].
4. The least of these values (outputs) is the minimum.
The greatest is the maximum.
**Make sure you give the y-value this is the extreme value!**
Critical Numbers
To find the max and min of f on [a, b]:
1. Make sure f is continuous on [a, b].
2. Find all critical numbers c1, c2, c3…cn of f
which are in (a, b) where f’(x) = 0 or f’(x) is
undefined.
3. Evaluate f(a), f(b), f(c1), f(c2), …f(cn).
4. The largest and smallest values in part 2 are the
max and min of f on [a, b].
Example
Example 4 Find all critical numbers
f ( x)  2 x 3  21x 2  60 x  4
Domain:
(–, +)
f ' ( x)  6 x 2  42 x  60  6( x  2)( x  5)
Critical number:
x = 2 and x = 5
Example
Example 5 Find all critical numbers.
Domain:
x2
f ( x) 
x 1
x ≠ 1, xR
2 x( x  1)  x 2 x 2  2 x x( x  2)


f ' ( x) 
2
2
( x  1)
( x  1)
( x  1)2
Critical number:
x = 1, x = 0, and x = 2
f ' (0)  f ' (2)  0, f ' (1) does not exist
Example
Example 6 Find all critical numbers.
Domain:
f ' ( x) 
(–, +)
2
33 x  4
Critical number:
x = –4
f ' (4) does not exist
f ( x )  ( x  4)
f’(–4 )
2
3
Example
Example 7 Find the max and min of f on the interval [–3, 5].
f ( x)  x3  3x 2  24 x  2
Domain:
(–, +)
f ' ( x)  3x 2  6 x  24  3( x  2)( x  4)
Critical number:
x = –2 and x = 4
Graph is not in scale
x
f (x)
Left Endpoint
f (–3)= 20
Critical Number
Critical Number
f (–2)= 30
f (4)=–78
maximum
minimum
Right Endpoint
f (5)=–68
Practice Of
Example 7 Find the extrema of f on the interval [–1, 2].
f ( x)  3 x 4  4 x 3
Domain:
(–, +)
f ' ( x)  12 x 3  12 x 2  12 x 2 ( x  1)
Critical number:
x
f (x)
Left Endpoint
f (–1)= 7
x = 0 and x = 1
Critical Number
f (0)= 0
Critical Number
f (1)=–1
minimum
Right Endpoint
f (2)=16
maximum
f ’(0) does not exist
Example
Example 8 Find the extrema of f on
the interval [–1, 3].
f ( x)  2 x  3x
2
3
 x 3 1
2
f ' ( x)  2  1  2 1 
 3 
3
x
 x 
Critical number:
x = 0 and x = 1
1
x
f (x)
Left Endpoint
f (–1)= –5
minimum
Critical Number
f (0)= 0
maximum
Critical Number
f (1)=–1
Right Endpoint
f (3)=
6  3 3 9  0.24
Practice Of
Example 8 Find the extrema of f on
the interval [0, 2].
f ( x)  2 sin x  cos 2 x
f ' ( x)  2 cos x  2 sin 2 x
 2 cos x  4 sin x cos x
 2 cos x(1  2 sin x)
Critical number:
x = /2, x = 3/2, x = 7/6, x = 11/6
x
f (x)
Left
Endpoint
f (0)=–1
Critical
Number
Critical
Number
Critical
Number
Critical
Number
f (/2)= f (7/6)
3
=–3/2
f (3/2)
=–1
f (11/6) f (2) =–
1
=–3/2
max
min
min
Right
Endpoint
Summary
Open vs. Closed Intervals
1. An open interval MAY have extrema
2. A closed interval on a continuous curve
will ALWAYS have a minimum and a
maximum value.
3. The min & max may be the same value
 How?
Homework
Section 3.1 page 169 #1,2,13-16,19-24,36,60