Deck 5: Applications of Derivatives

Question 1

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].

A) No absolute extrema.
B) Absolute maximum only.
C) Absolute minimum and absolute maximum.
D) Absolute minimum only.

Answer 1

B

Question 2

Solve the problem.

Answer 2

B

Question 3

Find the location of the indicated absolute extremum for the function.

A) x = -3
B) x = -5
C) x = 5
D) x = 3

Answer 3

D

Question 4

Find the location of the indicated absolute extremum for the function.

A) x = 2
B) x = 0
C) x = -4
D) x = -3

Question 5

Find the location of the indicated absolute extremum for the function.

Question 6

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].

A) No absolute extrema.
B) Absolute minimum only.
C) Absolute maximum only.
D) Absolute minimum and absolute maximum.

Question 7

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].

A) Absolute minimum and absolute maximum.
B) Absolute minimum only.
C) Absolute maximum only.
D) No absolute extrema.

Question 8

Solve the problem.

Question 9

Solve the problem.

Question 10

Find the location of the indicated absolute extremum for the function.

A) No maximum
B) x = -1
C) x = 4
D) x = 1

Question 11

Find the location of the indicated absolute extremum for the function.

A) x = 0
B) No maximum
C) x = 3
D) x = 5

Question 12

Find the location of the indicated absolute extremum for the function.

A) No minimum
B) x = -1
C) x = 2
D) x = 1

Question 13

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].

A) Absolute minimum only.
B) No absolute extrema.
C) Absolute minimum and absolute maximum.
D) Absolute maximum only.

Question 14

Find the location of the indicated absolute extremum for the function.

A) x = -4
B) x = 1
C) No maximum
D) x = 4

Question 15

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.

Question 16

Find the location of the indicated absolute extremum for the function.

A) No maximum
B) x = 0
C) x = -1
D) x = 2

Question 17

Solve the problem.

Question 18

Find the location of the indicated absolute extremum for the function.

A) x = -1
B) x = 1
C) x = 2
D) x = -2

Question 19

Sketch the graph of the function and determine whether it has any absolute extreme values on its domain.

Question 20

Determine from the graph whether the function has any absolute extreme values on the interval [a, b].

A) Absolute minimum only.
B) Absolute minimum and absolute maximum.
C) No absolute extrema.
D) Absolute maximum only.

Question 21

Find the absolute extreme values of the function on the interval.

Question 22

Find the absolute extreme values of the function on the interval.

A) absolute maximum is 405 at x = -27 ; absolute minimum is 80 at x = 8
B) absolute maximum is 80 at x = 8 ; absolute minimum is 0 at x = 0
C) absolute maximum is 81 at x = -27 ; absolute minimum is 0 at x = 0
D) absolute maximum is 405 at x = -27 ; absolute minimum is 0 at x = 0

Question 23

Find the absolute extreme values of the function on the interval.

Question 24

Determine all critical points for the function.

A) x = 0 and x = 1
B) x = 0 and x = 2
C) x = -1 and x = 1
D) x = 0

Question 25

Find the absolute extreme values of the function on the interval.

A) absolute maximum is 512 at x = 8; absolute minimum is 0 at x = 01
B) absolute maximum is 256 at x = 8; absolute minimum is 0 at x = 01
C) absolute maximum is 256 at x = 8; absolute minimum is 1 at x = -1
D) absolute maximum is 256 at x = 8; absolute minimum does not exist

Question 26

Determine all critical points for the function.

A) x = -12
B) x = -6
C) x = 6
D) x = 0

Question 27

Find the absolute extreme values of the function on the interval.

Question 28

Find the absolute extreme values of the function on the interval.

Question 29

Find the absolute extreme values of the function on the interval.

Question 30

Find the absolute extreme values of the function on the interval.

Question 31

Determine all critical points for the function.

A) x = -2, x = 0, and x = 2
B) x = 2
C) x = -2 and x = 2
D) x = -2

Question 32

Find the absolute extreme values of the function on the interval.

Question 33

Find the absolute extreme values of the function on the interval.

A) absolute maximum is 5 at x = 4; absolute minimum is - 9 at x = -3
B) absolute maximum is 11 at x = -4; absolute minimum is - 9 at x = 3
C) absolute maximum is 11 at x = 4; absolute minimum is - 3 at x = -3
D) absolute maximum is 5 at x = -3; absolute minimum is - 3 at x = 4

Question 34

Find the absolute extreme values of the function on the interval.

Question 35

Determine all critical points for the function.

A) x = -4
B) x = -4 and x = 4
C) x = 0, x = -4, and x = 4
D) x = 4

Question 36

Find the absolute extreme values of the function on the interval.

A) absolute maximum is 4 at x = 9; absolute minimum is 0 at x = 5
B) absolute maximum is 1 at x = 4; absolute minimum is 0 at x = 5
C) absolute maximum is -1 at x = 4; absolute minimum is 4 at x = 9
D) absolute maximum is 4 at x = 9; absolute minimum is 1 at x = 4

Question 37

Find the absolute extreme values of the function on the interval.

Question 38

Find the absolute extreme values of the function on the interval.

Question 39

Find the absolute extreme values of the function on the interval.

Question 40

Find the absolute extreme values of the function on the interval.

Question 41

Find the derivative at each critical point and determine the local extreme values.

Question 42

Find the extreme values of the function and where they occur.

A) The maximum value is 0 at x = 1.
B) There are no definable extrema.
C) The minimum value is 0 at x = -1.
D) The minimum value is 0 at x = 1.

Question 43

Find the derivative at each critical point and determine the local extreme values.

Question 44

Find the extreme values of the function and where they occur.

A) The minimum is -4 at x = -1.
B) The minimum is -1 at x = 4.
C) The minimum is 1 at x = -4.
D) The minimum is 1 at x = 4.

Question 45

Find the extreme values of the function and where they occur.

A) The maximum value is 1 at x = 0.5, the minimum value is -1 at x = 0.5.
B) The minimum value is -1 at x = 0.5.
C) The maximum value is 1 at x = 0.5.
D) The maximum value is 1 at x = 0.

Question 46

Find the extreme values of the function and where they occur.

A) The minimum value is 0 at x = 0.
B) The minimum value is 0 at x = 1. The maximum value is 0 at x = -1.
C) The maximum value is 0 at x = 0.
D) The minimum value is - 92 at x = -1. The maximum value is 92 at x = 1.

Question 47

Find the extreme values of the function and where they occur.

Question 48

Find the derivative at each critical point and determine the local extreme values.

Question 49

Find the extreme values of the function and where they occur.

Question 50

Determine all critical points for the function.

A) x = 0, x = 6, and x = 7
B) x = 6 and x = 7
C) x = 0 and x = 6
D) x = 6

Question 51

Find the extreme values of the function and where they occur.

Question 52

Find the derivative at each critical point and determine the local extreme values.

Question 53

Find the extreme values of the function and where they occur.

Question 54

Find the extreme values of the function and where they occur.

A) Local maximum at (0, 1).
B) None
C) Local minimum at (2, -3).
D) Local maximum at (0, 1), local minimum at (2, -3).

Question 55

Determine all critical points for the function.

A) x = -2
B) x = -12 and x = 0
C) x = 2
D) x = 0 and x = -2

Question 56

Find the extreme values of the function and where they occur.

A) Local maximum at (0, -1).
B) None
C) Local maximum at (1, 0), local minimum at (-1, 0).
D) Local maximum at (-1, 0), local minimum at (1,0).

Question 57

Find the extreme values of the function and where they occur.

A) None
B) Local maximum at (2, -14), local minimum at (-2, 18).
C) Local maximum at (-2, 18), local minimum at (2, -14).
D) Local maximum at (0, 0).

Question 58

Find the extreme values of the function and where they occur.

A) The minimum is 4 at x = -1.
B) None
C) The maximum is 4 at x = 1.
D) The maximum is 4 at x = 2.

Question 59

Find the extreme values of the function and where they occur.

A) The minimum is 0 at x = 1.
B) The maximum is 6 at x = 2.
C) The maximum is 6 at x = -2.
D) The minimum is 6 at x = 0.

Question 60

Determine all critical points for the function.

A) x = 0
B) x = 4
C) x = 0, x = 4, and x = -4
D) x = 0 and x = 4

Question 61

Answer the problem.

Question 62

Question 63

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.

Question 64

Answer the problem.
b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a
solution). You may wish to plot f as well. List the points as ordered pairs (x, y).
c). Find the interior points where f does not exist. List the points as ordered pairs (x, y).
d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y).
e). Find the function's absolute extreme values on the interval and identify where they occur.

Question 65

Find the derivative at each critical point and determine the local extreme values.

Question 66

Answer the problem.

Question 67

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.

Question 68

Question 69

Answer the problem.
Imagine there is a function for which f(x) = 0 for all x. Does such a function exist? Is it reasonable to say that all
values of x are critical points for such a function? Is it reasonable to say that all values of x are extreme values
for such a function. Give reasons for your answer.

Question 70

Answer the problem.

Question 71

Answer the problem.

A) 8 hundred thousand
B) 10 hundred thousand
C) 3 hundred thousand
D) 5 hundred thousand

Question 72

Answer the problem.
b). Find the interior points where f = 0 (you may need to use the numerical equation solver to approximate a
solution). You may wish to plot f as well. List the points as ordered pairs (x, y).
c). Find the interior points where f does not exist. List the points as ordered pairs (x, y).
d). Evaluate the function at the endpoints and list these points as ordered pairs (x, y).
e). Find the function's absolute extreme values on the interval and identify where they occur.

Question 73

Answer the problem.

Question 74

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.

Question 75

Answer the problem.
If the derivative of an even function f(x) is zero at x = c, can anything be said about the value of f at x = -c? Give
reasons for your answer.

Question 76

Graph the function, then find the extreme values of the function on the interval and indicate where they occur.

Question 77

Find the derivative at each critical point and determine the local extreme values.

Question 78

Find the derivative at each critical point and determine the local extreme values.

Question 79

Question 80

Answer the problem.
If the derivative of an odd function g(x) is zero at x = c, can anything be said about the value of g at x = -c?
Give reasons for you answer.