Deck 4: Applications of the Derivative

Question 1

Determine all critical points for the function.

-f(x) = + 6x + 9

A) x = 3
B) x = 0
C) x = -6
D) x = -3

Answer 1

x = -3

Question 2

Determine all critical points for the function.

-f(x) = - 12x - 5

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

Answer 2

x = -2 and x = 2

Question 3

Determine all critical points for the function.

-f(x) = - 9 + 10

A) x = 0 and x = 6
B) x = -3 and x = 3
C) x = 0
D) x = 0 and x = 3

Answer 3

x = 0 and x = 6

Question 4

Determine all critical points for the function.

-f(x) = 5 - 3

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

Question 5

Determine all critical points for the function.

-f(x) =

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

Question 6

Determine all critical points for the function.

-f(x) =

A) x = 0 and x = 1
B) x = 1
C) x = 1 and x = 7
D) x = 0, x = 1, and x = 7

Question 7

Determine all critical points for the function.

-y = 3 - 96

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

Question 8

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

-g(x) = - + 5x - 6, 2 $\le$ x $\le$ 3

A) absolute maximum is 1/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2
B) absolute maximum is 5/4 at x = 7/2; absolute minimum is 0 at 3 and 0 at x = 2
C) absolute maximum is 49/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2
D) absolute maximum is 1/4 at x = 5/2; absolute minimum is 0 at 3 and 0 at x = 2

Question 9

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

-f( $\theta$ ) = sin , 0 $\le$ $\theta$ $\le$

A) absolute maximum is 1 at $\theta$ = 0; absolute minimum is -1 at $\theta$ = $\pi$
B) absolute maximum is 1 at $\theta$ = 7/8 $\pi$ ; absolute minimum is -1 at $\theta$ = 1/8 $\pi$ ,
C) absolute maximum is 1 at $\theta$ = 9/8 $\pi$ ; absolute minimum is -1 at $\theta$ = 7/8 $\pi$
D) absolute maximum is 1 at $\theta$ = 1/8 $\pi$ ; absolute minimum is -1 at $\theta$ = 7/8 $\pi$ ,

Question 10

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

-f(x) = csc x, - $\le$ x $\le$

A) absolute maximum is 1 at x = $\pi$ ; absolute minimum is -1 at x = $\pi$
B) absolute maximum does not exist; absolute minimum does not exist
C) absolute maximum is 0 at x = - $\pi$ ; absolute minimum is -1 at x = $\pi$
D) absolute maximum is -1 at x = $\pi$ ; absolute minimum is 1 at x = 0

Question 11

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

-F(x) = - , 0.5 $\le$ x $\le$ 2

A) absolute maximum is 1/2 at x = 1/ 2; absolute minimum is -8 at x = 2
B) absolute maximum is - 1/2 at x = 2 ; absolute minimum is -8 at x = 1/2
C) absolute maximum is - 1/2 at x = 1/ 2; absolute minimum is -8 at x = - 2
D) absolute maximum is at x = -1/2 at x = 2 ; absolute minimum is -8 at x = -1/2

Question 12

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

-F(x) = , -1 $\le$ x $\le$ 27

A) absolute maximum is 3 at x = 27; absolute minimum is 0 at x =0
B) absolute maximum is 3 at x = 27; absolute minimum is -3 at x = -27
C) absolute maximum is 0 at x = 0; absolute minimum is 3 at x = 27
D) absolute maximum is 3 at x = -27; absolute minimum is 0 at x =0

Question 13

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

-h(x) = x + 5, -2 $\le$ x $\le$ 3

A) absolute maximum is - 7/2 at x = -2; absolute minimum is 4 at x = 3
B) absolute maximum is 13/2 at x = 3; absolute minimum is 4 at x = -2
C) absolute maximum is - 7/2 at x = -3; absolute minimum is -3 at x = 2
D) absolute maximum is - 7/2 at x = 3; absolute minimum is 4 at x = -2

Question 14

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

-g(x) = 7 - 6 , -2 $\le$ x $\le$ 3

A) absolute maximum is 14 at x = 0; absolute minimum is -17 at x = 3
B) absolute maximum is 7 at x = 0; absolute minimum is -47 at x = 3
C) absolute maximum is 6 at x = 0; absolute minimum is -61 at x = 3
D) absolute maximum is 42 at x = 0; absolute minimum is -17 at x = -2

Question 15

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

-f(x) = tan x, - $\le$ x $\le$

A)
B)
C)
D)

Question 16

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

-f(x) = , -1 $\le$ x $\le$ 8

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

Question 17

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

-f(x) = 7 , -27 $\le$ x $\le$ 8

A) absolute maximum is 1792 at x = 8 ; absolute minimum is 0 at x = 0
B) absolute maximum is 6561 at x = -27 ; absolute minimum is 0 at x = 0
C) absolute maximum is 45,927 at x = -27 ; absolute minimum is 0 at x = 0
D) absolute maximum is 45,927 at x = -27 ; absolute minimum is 1792 at x = 8

Question 18

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

-f(x) = ln(x + 2) + , 1 $\le$ x $\le$ 5

A) absolute minimum value is ln 4 + 1/2 at x = 2; absolute maximum value is ln 3 + 1 at x = 1
B) absolute minimum value is ln 4 + 1/2 at x = 2; absolute maximum value is ln 7 + 1/5 at x = 5
C) absolute minimum value is ln 3 + 1 at x = 1; absolute maximum value is ln 7 + 1/5 at x = 5
D) absolute minimum value is -1 at x = -1; absolute maximum value is ln 7 + 1/5 at x = 5

Question 19

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

-f(x) = - 6 , - $\infty$ < x < $\infty$

A) no minimum value and no maximum value
B) absolute maximum value is - 6 at x = 0; no minimum value
C) absolute minimum value is - 6 at x = 0; no maximum value
D) absolute minimum value is - 6 at x = 0; absolute maximum value is - 6/e at x = 1

Question 20

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

-f(x) = ln (-x), -6 $\le$ x $\le$ -1

A) absolute maximum value is 0 at x = -1; absolute minimum value is -ln 6 at x = -6
B) absolute minimum value is 0 at x = -1; no maximum value
C) no minimum value; no maximum value
D) absolute minimum value is 0 at x = -1; absolute maximum value is ln 6 at x = -6

Question 21

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

-f(x) = - x, -4 $\le$ x $\le$ 2

A) absolute minimum value is + 4 at x = -4; absolute maximum value is - 2 at x = 2
B) absolute minimum value is 1 at x = 0; absolute maximum value is - 2 at x = 2
C) absolute minimum value is 1 at x = 0; absolute maximum value is + 4 at x = -4
D) absolute minimum value is 1 at x = 0; no maximum value

Question 22

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

-y = x² + 2x - 3

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

Question 23

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

-y = x³ - 3x² + 1

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

Question 24

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

-y = x³ - 12x + 2

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

Question 25

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

-y =

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

Question 26

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

-y =

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

Question 27

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

-y =

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

Question 28

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

-y = - 3 + 6x - 8

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

Question 29

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

-y =

A) Absolute minimum is 0 at x = 1.
B) Absolute maximum is 10 at x = 2.
C) Absolute minimum is 10 at x = 0.
D) Absolute maximum is 10 at x = -2.

Question 30

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

-y =

A) None
B) Absolute maximum is 1/3 at x = 0; absolute minimum is - 1 at x = -2.
C) Absolute maximum is 3 at x = 0; absolute minimum is 1/3 at x = -2.
D) Absolute maximum is -1/3 at x = 0; absolute minimum is 1 at x = -2.

Question 31

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

-y =

A) Absolute minimum value is 0 at x = 0; no maximum value.
B)Absolute minimum value is 0 at x = 0, absolute maximum value is 4 at x = -2.
C) Absolute minimum value is 4 at x = -2; no maximum value.
D) None

Question 32

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

-y =

A) Absolute minimum value is at x = ; no maximum value.
B) Absolute maximum value is at x = ; no minimum value.
C) Absolute maximum value is at x = ; absolute minimum value is 0 at x = 1.
D) None

Question 33

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

-y = + 2x

A)
B)
C)
D) None

Question 34

Provide an appropriate response.
-Imagine there is a function for which (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 35

Provide an appropriate response.
-Consider the quartic function f(x) = a + b + c + dx + e, a ≠ 0. Must this function have at least one critical point? Give reasons for your answer. (Hint: Must for some x?) How many local extreme values can f have?

Question 36

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval.

-f(x) = ,

Question 37

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval.

-g(x) = ,

Question 38

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval.

-s(t) = ,

Question 39

Find the value or values of c that satisfy the equation = (c) in the conclusion of the Mean Value Theorem for the function and interval.

-f(x) = + 2x + 1, [ -3, -2]

A) -3, -2
B) - 5/2, 5/2
C) 0, - 5/2
D) - 5/2

Question 40

Find the value or values of c that satisfy the equation = (c) in the conclusion of the Mean Value Theorem for the function and interval.

-f(x) = x + ,

A) 0, 2
B) -2 , 2
C) 3, 4
D) 2

Question 41

Find the value or values of c that satisfy the equation = (c) in the conclusion of the Mean Value Theorem for the function and interval.

-f(x) = x, [-1, 1]

A) c = 0,
B) c = - ,
C) c =
D) c = - , 0 ,

Question 42

Find the value or values of c that satisfy the equation = (c) in the conclusion of the Mean Value Theorem for the function and interval.

-f(x) = ln (x - 3), [ 4, 8]

A) c = + 3
B) c = + 3
C) c = + 3
D) c = + 3

Question 43

Provide an appropriate response.
-It took 20 seconds for the temperature to rise from 4° F to 166° F when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some time, the temperature was increasing at a rate of ° F/sec. Explain.

Question 44

Solve the problem.

-Select an appropriate graph of a twice-differentiable function y = f(x) that passes through the points and whose first two derivatives have the following sign patterns.

A)
B)
C)
D)

Question 45

Write the word or phrase that best completes each statement or answers the question.

-Sketch a continuous curve y = f(x) with the following properties: f(2) = 3; (x) > 0 for x > 4; and (x) < 0 for x < 4 .

Question 46

Choose the one alternative that best completes the statement or answers the question.

-The graph below shows the first derivative of a function y = f(x). Select a possible graph of f that passes through the point P.

A)
B)
C)
D)

Question 47

Choose the one alternative that best completes the statement or answers the question.

-The graph below shows the first derivative of a function y = f(x). Select a possible graph f that passes through the point P.

A)
B)
C)
D)

Question 48

Choose the one alternative that best completes the statement or answers the question.

-The graph below shows the first derivative of a function . Select a possible graph f that passes through the point P.

A)
B)
C)
D)

Question 49

Find the largest open interval where the function is changing as requested.

-Increasing f(x) = x² - x

A) (-1, 1)
B) (- $\infty$ , -1)
C) (- $\infty$ , $\infty$ )
D) (1, $\infty$ )

Question 50

Find the largest open interval where the function is changing as requested.

-Increasing f(x) = x² - 2x + 1

A) (0, $\infty$ )
B) (1, $\infty$ )
C) (- $\infty$ , 1)
D) (- $\infty$ , 0)

Question 51

Find the largest open interval where the function is changing as requested.

-Increasing y = (x² - 9)²

A) (-3, 0)
B) (- $\infty$ , 0)
C) (-3, 3)
D) (3, $\infty$ )

Question 52

Find the largest open interval where the function is changing as requested.

-Increasing f(x) =

A) (1, $\infty$ )
B) (- $\infty$ , 0)
C) (- $\infty$ , 1)
D) (0, $\infty$ )

Question 53

Find the largest open interval where the function is changing as requested.

-Decreasing f(x) =

A) (- $\infty$ , -4)
B) (4, $\infty$ )
C) (- $\infty$ , 4)
D) (-4, $\infty$ )

Question 54

Find the largest open interval where the function is changing as requested.

-Decreasing f(x) =

A) (- $\infty$ , -8)
B) (8, $\infty$ )
C) (- $\infty$ , 8)
D) (-8, $\infty$ )

Question 55

Find the largest open interval where the function is changing as requested.

-Decreasing y = + 7

A) (-7, 0)
B) (0, $\infty$ )
C) (7, $\infty$ )
D) (-7, 7)

Question 56

Find the largest open interval where the function is changing as requested.

-Decreasing f(x) = -

A) (3, $\infty$ )
B) (-3, $\infty$ )
C) (- $\infty$ , 3)
D) (- $\infty$ , -3)

Question 57

Find the largest open interval where the function is changing as requested.

-Decreasing f(x) = x³ - 4x

A)
B)
C)
D)

Question 58

Solve the problem.

-The graphs below show the first and second derivatives of a function . Select a possible graph f that passes through the point P.

A)
B)
C)
D)

Question 59

Solve the problem.

-The graphs below show the first and second derivatives of a function . Select a possible graph of f that passes through point P.

A)
B)
C)
D)

Question 60

Solve the problem.

-The graphs below show the first and second derivatives of a function . Select a possible graph f that passes through the point P.

A)
B)
C)
D)

Question 61

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.

-

A) Local minimum at x = 1; local maximum at x = -1; concave down on (0, $\infty$ ); concave up on (- $\infty$ , 0)
B) Local minimum at x = 1; local maximum at x = -1; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)
C) Local minimum at x = 1; local maximum at x = -1; concave down on (- $\infty$ , $\infty$ )
D) Local minimum at x = 1; local maximum at x = -1; concave up on (- $\infty$ , $\infty$ )

Question 62

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.

-

A) Local minimum at x = 1; local maximum at x =-1; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)
B)Local minimum at x = 1; local maximum at x =-1; concave down on (0, $\infty$ ); concave up on (- $\infty$ 0)
C) Local maximum at x = 1; local minimum at x =-1; concave up on (- $\infty$ , $\infty$ )
D) Local maximum at x = 1; local minimum at x =-1; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)

Question 63

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.

-

A) Local minimum at x = 3; local maximum at x = -3 ; concave down on (0, $\infty$ ); concave up on (- $\infty$ , 0)
B) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, -3) and (3, $\infty$ ); concave down on (-3, 3)
C) Local minimum at x = 3; local maximum at x = -3 ; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)
D) Local maximum at x = 3; local minimum at x = -3 ; concave up on (0, -3) and (3, $\infty$ ); concave down on (-3, 3)

Question 64

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.

-

A) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (- $\infty$ , -3) and (3, $\infty$ ); concave down on (-3, 3)
B) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (- $\infty$ -3) and (3, $\infty$ ); concave up on (-3, 3)
C) Local minimum at x = 3 ; local maximum at x = -3 ; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)
D) Local minimum at x = 3 ; local maximum at x = -3 ; concave down on (0, $\infty$ ); concave up on (- $\infty$ , 0)

Question 65

Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down.

-

A) Local minimum at x = 0; local maximum at x = 2; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)
B) Local minimum at x = 0; local maximum at x = 2; concave down on (0, $\infty$ ); concave up on (- $\infty$ , 0)
C) Local minimum at x = 2; local maximum at x = 0; concave down on (0, $\infty$ ); concave up on (- $\infty$ , 0)
D) Local minimum at x = 2; local maximum at x = 0; concave up on (0, $\infty$ ); concave down on (- $\infty$ , 0)

Question 66

Solve the problem.

-Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.

A)
B)
C)
D)

Question 67

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y = 3x² + 24x

A) local minimum: ( 8, -24) no inflection points

B) local minimum: ( -8, -24) no inflection points

C)local minimum: ( -4, -48) no inflection points

D) local minimum: ( 4, -48) no inflection points

Question 68

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y =

A)
B)
C)
D)

Question 69

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y = 2x³ - 15x² + 24x

A)
B)
C)

D)

Question 70

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y = x^1/3(x² - 63)

A)
B)

C)

D)

Question 71

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y =

A)

B)

C)

D)

Question 72

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y = x + cos 2x, 0 $\le$ x $\le$ $\infty$

A)
B)
C)
D)

Question 73

Graph the equation. Include the coordinates of any local extreme points and inflection points.

-y = x

A)
B)
C)
D)

Question 74

Sketch the graph and show all local extrema and inflection points.

-y = - + 4 - 2

A)
B)
C)
D)

Question 75

Sketch the graph and show all local extrema and inflection points.

-y = x

A)

B)

C)

D)

Question 76

Sketch the graph and show all local extrema and inflection points.

-y = x + sin x, 0 $\le$ x $\le$ 2 $\pi$

A)

B)

C)

D)

Question 77

Sketch the graph and show all local extrema and inflection points.

-y = | - 4x|

A)

B)

C)

D)

Question 78

Sketch the graph and show all local extrema and inflection points.

-y = ln ( 7 - )

A)

B)

C)

D)

Question 79

Sketch the graph and show all local extrema and inflection points.

-y = - 6 - 7x

A)
B)

C)

D)

Question 80

For the given expression , find y'' and sketch the general shape of the graph of y = f(x).

-y' = - 1

A)

B)

C)

D)