5.2 Relative Extrema
•
Find Relative Extrema of a function using the
first derivative
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Inc. Publishing as Pearson AddisonWesley
Relative Extrema
If a function f has a relative extreme value f
(c) on an open interval; then c is a critical
value. So,
f (c) = 0 or f (c) does not exist.
Relative extrema will only occur at points
where the derivative is = 0 or where it is
undefined.
Relative Minimum
Let I be the domain of f :
f (c) is a relative minimum (bottom of a valley) if
there exists within I an open interval I1 containing
c such that f (c) ≤ f (x) for all x in I1;
f has a relative minimum at c if f (x) < 0 on
(a, c) and f (x) > 0 on (c, b). That is, f is
decreasing to the left of c and increasing to the
right of c.
If the graph is continuous (no break) at
the point where the function changes
from decreasing to increasing, that point
is called a relative minimum point
Relative Maximum
Let I be the domain of f :
F (c) is a relative maximum (top of a hill) if
there exists within I an open interval I2
containing c such that f (c) ≥ f (x) for all x in I2.
f has a relative maximum at c if f (x) > 0 on
(a, c) and f (x) < 0 on (c, b). That is, f is
increasing to the left of c and decreasing to the
right of c.
If the graph is continuous (no break) at
the point where the function changes
from increasing to decreasing, that point
is called a relative maximum point.
Using First Derivatives to Find Maximum
and Minimum Values
Graph over the
interval (a, b)
a
a
c
c
f (c)
Sign of f (x) Sign of f (x)
for x in (a, c) for x in (c, b)
Increasing or
decreasing
Relative
minimum
–
+
Decreasing
on (a, c];
increasing on
[c, b)
Relative
maximum
+
–
Increasing on
(a, c];
decreasing
on [c, b)
b
b
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 7
Using First Derivatives to Find Maximum
and Minimum Values
Graph over the
interval (a, b)
f (c)
No
relative
maxima
or
minima
a
c
c
–
–
Increasing or
decreasing
Decreasing
on (a, b)
Increasing on
(a, b)
b
No
relative
maxima
or
minima
a
Sign of f (x) Sign of f (x)
for x in (a, c) for x in (c, b)
+
+
b
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Slide 2.1- 8
Using First Derivatives to Find Maximum
and Minimum Values
Example 1: For the function f given by
f (x)  2x 3  3x 2  12x  12.
find the relative extrema.
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 9
Using First Derivatives to Find Maximum
and Minimum Values
Example 1 (continued): Find Derivative
And set it = 0
6x 2  6x  12  0
x2  x  2
(x  2)(x  1)
x2
or
 0
 0
x  1
These two critical values partition the number line into
3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).
A
B
-1
C
2
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Slide 2.1- 10
Using First Derivatives to Find Maximum
and Minimum Values
Example 1 (continued):
3rd analyze the sign of f (x) in each interval.
Interval
A
C
B
-1
x
2
Test Value
x = –2
x=0
x=4
Sign of
f (x)
+
–
+
Result
f is increasing f is decreasing on
on (–∞, –1]
[–1, 2]
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f is increasing
on [2, ∞)
Slide 2.1- 11
Using First Derivatives to Find Maximum
and Minimum Values
Example 1 (concluded):
Therefore, by the First-Derivative Test,
f has a relative maximum at x = –1 given by
f (1)  2(1)3  3(1)2  12(1)  12  19
The relative maximum value is 19. It occurs where x = -1.
And f has a relative minimum at x = 2 given by
f (2)  2(2)3  3(2)2  12(2)  12  8
The relative minimum value is -8. It occurs where x is 2.
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Slide 2.1- 12
Using First Derivatives to Find Maximum
and Minimum Values
Example 1 (continued):
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Slide 2.1- 13
Using First Derivatives to Find Maximum
and Minimum Values
Example 3: Find the relative extrema for the
Function f (x) given by
f (x)  (x  2)2 3  1
Then sketch the graph.
1st find f (x).
1 3
2
f (x)  x  2 
3
2
f (x)  3
3 x2
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Slide 2.1- 14
Using First Derivatives to Find Maximum
and Minimum Values
Example 3 (continued):
2nd find where f (x) does not exist or where f (x) = 0.
Note that f (x) does not exist where the denominator
equals 0. Since the denominator equals 0 when x = 2,
x = 2 is a critical value.
f (x) = 0 where the numerator equals 0. Since 2 ≠ 0,
f (x) = 0 has no solution.
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Thus, x = 2 is theCopyright
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critical
value.
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Slide 2.1- 15
Using First Derivatives to Find Maximum
and Minimum Values
Example 3 (continued):
3rd x = 2 partitions the number line into 2 intervals:
A (– ∞, 2) and B (2, ∞). So, analyze the signs of
f (x) in both intervals.
B
A
Interval
x
2
Test Value
Sign of f (x)
Result
x=0
x=3
–
+
f is decreasing f is increasing on
on
(–Pearson
∞, 2]
[2, ∞)
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2008
Education,
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Slide 2.1- 16
Using First Derivatives to Find Maximum
and Minimum Values
Example 3 (continued):
Therefore, by the First-Derivative Test,
f has a relative minimum at x = 2 given by
f (2)  (2  2)2 3  1  1
The relative minimum value is 1. It occurs at x = 2.
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 17
Using First Derivatives to Find Maximum
and Minimum Values
Example 3 (concluded):
We use the information obtained to sketch the
graph below, plotting other function values as
needed.
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 18
More Examples
Find the relative extrema.
f ( x)  x  1
3
x
f ( x) 
x4
2
x  4 x  21
f ( x) 
x2
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 19
C(q)  25q  5000
p(q)  90  0.02 q
C(q) above is the cost function. p(q) is the price function.
Find
a) The number of units that will produce a maximum profit.
b) The maximum profit.
c) The price that will produce a maximum profit.
a) 1625 b) $47812.50 c) $57.50
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Inc. Publishing as Pearson AddisonWesley
Slide 2.1- 20