Definition: critical numbermayaj/Chapter5_Sec5P1completed.pdfSection 5.1 The First Derivative...
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Section 5.1 The First Derivative Increasing/Decreasing Test: (a) If f 0 (x) > 0 on an interval, then f is increasing on that interval (b) If f 0 (x) < 0 on an interval, then f is decreasing on that interval Definition: A critical number of a function f is a number c in the domain of f such that either f 0 (c) = 0 or f 0 (c) does not exist. 1. Find the critical numbers for f (x)= x 2 - 6x. Steps for Determining where f is Increasing/Decreasing 1. Find all critical numbers. 2. Plot critical numbers on a number line. 3. Choose values to test regions on the number line around the critical numbers. 4. Plug test values into f 0 and record the sign. If the sign is positive (+) then f is increasing on that interval, and if the sign is negative (-) then f is decreasing on that interval. 2. Find where the function in example 1 is increasing and decreasing.
Transcript of Definition: critical numbermayaj/Chapter5_Sec5P1completed.pdfSection 5.1 The First Derivative...
Section 5.1 The First Derivative
Increasing/Decreasing Test:
(a) If f
0(x) > 0 on an interval, then f is increasing on that interval
(b) If f
0(x) < 0 on an interval, then f is decreasing on that interval
Definition: A critical number of a function f is a number c in the
domain of f such that either f
0(c) = 0 or f
0(c) does not exist.
1. Find the critical numbers for f(x) = x
2 � 6x.
Steps for Determining where f is Increasing/Decreasing
1. Find all critical numbers.
2. Plot critical numbers on a number line.
3. Choose values to test regions on the number line around the critical numbers.
4. Plug test values into f
0and record the sign. If the sign is positive (+) then f is increasing
on that interval, and if the sign is negative (-) then f is decreasing on that interval.
2. Find where the function in example 1 is increasing and decreasing.
Definition: Let c be a number in the domain D of a function f . Then
f(c) is the
Relative Maximum value of f if f(c) � f(x) when x is near c.
Relative Minimum value of f if f(c) f(x) when x is near c.
Note: We say that f(c) is a relative extremum if f(c) is either a relative maximum or a relative
minimum.
The First Derivative Test: Suppose that c is a critical number of a
continuous function f .
(a) If f
0(x) changes from positive to negative around c, then f has a rela-
tive maximum at c.
(b) If f
0(x) changes from negative to positive around c, then f has a rela-
tive minimum at c.
(c) If f
0(x) does not change sign around c (for example, f
0(x) is positive
on both sides of c or negative on both sides), then f has no relative
maximum or minimum at c.
3. For the function below, determine each of the following.
f(x) = 6x
3+ 9x
2 � 360x.
(a) Find all critical values of f(x).
(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
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(c) Find the x-coordinates of all relative extrema on the graph of f(x).
4. For the function below, determine each of the following.
f(x) =
x
2
x� 15
(a) Find all critical values of f(x).
(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
5. For the function below, determine each of the following.
f(x) = x
5e
�0.5x
(a) Find all critical values of f(x).
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ftp.5x4eo#tx5e&5xfo.s)=osYF+%e0*Tfotx4jo5xl5- 0.5×1=0
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(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
6. For the function defined below, determine each of the following.
f(x) = 5(x
2 � 36x+ 180)
4/5
(a) Find all critical values of f(x).
(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
4 Fall 2016, Maya Johnson
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7. Suppose the function f(x) has a domain of all real numbers and its derivative is shown below.
f
0(x) = (x+ 2)
2(x� 7)
4(x� 8)
7.
(a) Find all intervals on which f(x) is increasing.
(b) Find all intervals on which f(x) is decreasing.
8. Suppose the function f(x) has a domain of all real numbers except x = �7. The first derivative
of f(x) is shown below.
f
0(x) =
�5(x� 1)
(x+ 7)
7
(a) Find all intervals on which f(x) is increasing.
(b) Find all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
5 Fall 2016, Maya Johnson
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max
Dec.to/-7)UliTMaxatxJ
No min . since -7 is not in domain
9. Use the graph of f
0(x) to answer questions about the function f(x).
This is the graph of f
0(x).
(a) Find all critical values of f(x).
(b) Find all intervals on which f(x) is increasing and all intervals on which f(x) is decreasing.
(c) Find the x-coordinates of all relative extrema on the graph of f(x).
6 Fall 2016, Maya Johnson
10. Use the graph of the function f(x) displayed below to answer the following questions.
(a) Find the critical values where f
0(x) does not exist.
(b) Find the critical values where f
0(x) = 0.
(c) Find the x-coordinate(s) of the relative maxima for f(x).
(d) Find the x-coordinate(s) of the relative minima for f(x).
7 Fall 2016, Maya Johnson
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