Deck 3: Additional Applications of the Derivative

Question 1

Find all intervals where the derivative of the function shown below is negative.

Answer 1

2 < x < 3

Question 2

The derivative of function shown below is negative on the interval shown.

Answer 2

False

Question 3

Find all intervals where the derivative of the function shown below is negative.

Answer 3

-5 < x < 0 and 0 < x < 3

Question 4

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 5 x - 3$ .

A) $\text { Decreasing for } x < - \frac { 5 } { 2 } \text {; increasing for } x > - \frac { 5 } { 2 }$
B) $\text { Decreasing for } x > - \frac { 5 } { 2 } \text {; increasing for } x < - \frac { 5 } { 2 }$
C) Decreasing for all x
D) Increasing for all x

Question 5

Find the intervals of increase and decrease for the function $f ( x ) = x ^ { 2 } + 9 x - 4$ .

A) $\text { Decreasing for } x > - \frac { 9 } { 2 } \text { and increasing for } x < - \frac { 9 } { 2 }$
B) $\text { Decreasing for } x < - \frac { 9 } { 2 } \text { and increasing for } x > - \frac { 9 } { 2 }$
C) Decreasing for all x
D) Increasing for all x

Question 6

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 2$ .

A) Increasing on x $\le$ -2 and x $\ge$ 3, decreasing on - 2 $\le$ x $\le$ 3
B) Increasing on x < -3 and x > 2, decreasing on -3 < x < 2
C) Increasing on -3 < x < 2, decreasing on x < -3 and x > 2
D) Increasing on x < -2, decreasing on x > 3

Question 7

Find the intervals of increase and decrease for the function $f ( x ) = - x ^ { 5 } - x ^ { 3 } - 4 x + 10$ .

Question 8

Find the intervals of increase and decrease for the function $f ( x ) = - x ^ { 7 } - x ^ { 5 } - 3 x + 15$ .

Question 9

Find the intervals of increase and decrease for $f ( x ) = \frac { 8 x - 5 } { - 2 x + 10 }$ . Round numbers to two decimal places, if necessary.

A) Increasing on x < 5, decreasing on x > 5
B) Increasing on x < 5 and x > 5
C) Increasing on x $\le$ 0.63 and on x > 5, decreasing on 0.63 < x $\le$ 5
D) Increasing on 0.63 < x $\le$ 5, decreasing on x $\le$ 0.63 and on x > 5

Question 10

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Question 11

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 6 } + 8 }$ .

Question 12

Find the intervals of increase and decrease for the function $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }$ .

Question 13

Find all critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 4 }$ .

Question 14

Find all the critical numbers of the function $f ( x ) = - 2 x ^ { 4 } + 4 x ^ { 2 } + 1$ .

A) 0, 1, -1
B) -1
C) $\frac { 1 } { 2 }$
D) None

Question 15

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$

A) (0, 4) relative minimum; (3, 166) neither
B) (0, 2) neither; (1, 3) relative minimum
C) (0, 2) relative minimum; (1, 4) neither
D) (0, 2) relative minimum; (1, 3) relative maximum

Question 16

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 12 x - 5$ .

A) None
B) -2, 2
C) 0, -2, 2
D) $3 \sqrt { 5 }$

Question 17

Find all the critical numbers of the function $f ( x ) = 2 x ^ { 2 } - 8 x + 7$ .

A) -7
B) $- \frac { 7 } { 2 }$
C) 2
D) None

Question 18

Find all the critical numbers of the function. $f ( x ) = 4 x ^ { 2 } - 3 x + 1$

A) -1
B) $\frac { 3 } { 8 }$
C) $- \frac { 1 } { 4 }$
D) None

Question 19

Find all the critical numbers of the function $f ( x ) = x ^ { 3 } - 3 x - 2$ .

A) None
B) -1, 1
C) 0, -1, 1
D) $- \sqrt [ 3 ] { 2 }$

Question 20

Find all critical numbers of the function $f ( x ) = x ^ { 5 } + 5 x ^ { 6 }$ .

Question 21

Find all critical numbers of the function $f ( t ) = \sqrt { t ^ { 2 } - 4 t + 8 }$ .

Question 22

Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. Round numbers to two decimal places, if necessary. $$f ( x ) = \frac { 2 } { x ^ { 2 } - 8 x + 7 }$$

A) x = 1, 4, 7; (4, -0.22) relative maximum
B) x = 1, 4, 7; (1, 2) relative maximum; (4, -0.22) relative minimum; (7, 2) relative maximum
C) x = 1, 7; (1, 2) relative maximum; (7, 2) relative maximum
D) x = 1, 7; (1, 2) relative maximum; (7, 2) relative minimum

Question 23

Find all critical numbers of the function $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }$ .

Question 24

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 48 x - x ^ { 2 } } { x ^ { 2 } + 48 }$ million dollars. What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places, if necessary.

A) Maximum at x = 6 and maximum revenue is R(6) = 18 (million dollars)
B) Maximum at x = 6 and maximum revenue is R(6) = 2.05 (thousand dollars)
C) Maximum at x = 6 and maximum revenue is R(6) = 3 (million dollars)
D) Maximum at x = 7 and maximum revenue is R(7) = 2.05 (million dollars)

Question 25

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).

A) $a = - \frac { 4 } { 5 }$ , b = -8, c = -5
B) $a = - \frac { 4 } { 5 }$ , b = 8, c = 5
C) $a = \frac { 4 } { 5 }$ , b = 8, c = 5
D) $a = \frac { 4 } { 5 }$ , b = -8, c = 5

Question 26

The function graphed below has a positive second derivative everywhere.

Question 27

Determine where the second derivative in the function graphed below is positive.

Question 28

The function graphed below has a negative second derivative everywhere.

Question 29

Determine where the second derivative in the function graphed below is positive.

Question 30

Determine where the graph of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ is concave up and concave down.

Question 31

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Question 32

Determine where the graph of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 9 x + 1$ is concave down.

A) x > 1
B) x < 1
C) x > -1
D) x < -1

Question 33

Determine where the graph of $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 6 x + 3$ is concave down.

A) For x < 3
B) For x > 3
C) For x < -3
D) For x > -3

Question 34

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 7 }$ is concave up and concave down.

Question 35

The graph of $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 5$ is concave down for $x < \frac { 1 } { 2 }$ .

Question 36

The inflection point of $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } - 13$ is (-2, 3).

Question 37

Determine where the graph of $f ( x ) = x ^ { 4 } - 6 x ^ { 2 }$ is concave up.

A) for x < -1 and x > 1
B) for -1 < x < 1
C) Everywhere
D) Nowhere

Question 38

Locate all inflection points of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ .

A) (1, 9) and (-4, -486)
B) (1, 9)
C) None
D) (0, 26)

Question 39

Determine where the graph of $f ( x ) = x ^ { 4 } - 6 x ^ { 2 } + 2$ is concave up.

A) For x < -1 and x > 1
B) For -1 < x < 1
C) Everywhere
D) Nowhere

Question 40

The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x < \frac { 2 } { 3 }$ .

Question 41

The graph of $g ( t ) = t ^ { 4 } + 2 t ^ { 3 }$ is concave up everywhere.

Question 42

Determine where the graph of $g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }$ is concave up and concave down.

Question 43

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

A) $\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$
B) $( 0,0 ) , \left( 1 , \frac { 1 } { 2 } \right) , \left( - 1 , - \frac { 1 } { 2 } \right)$
C) (0, 0)
D) $( 0,0 ) , \left( - 1 , - \frac { 1 } { 2 } \right)$

Question 44

The function $f ( x ) = x ^ { 6 } - 5 x ^ { 3 } + 2$ has a relative maximum at x = 1.

Question 45

The function $f ( x ) = x ^ { 2 } + \frac { 2 } { x }$ has a relative minimum at x = 1.

Question 46

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Question 47

The second derivative test reveals that $f ( x ) = x ^ { 2 } - 3$ has

A) a relative maximum at $x = \sqrt { 3 }$
B) a point of inflection at $x = \sqrt { 3 }$
C) a relative minimum at x = 0.
D) a point of inflection at x = 3.

Question 48

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 4 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1$ .

A) Relative maximum at (1, 109); relative minimum at (-3, -19)
B) Relative maximum at (-3, -19); relative minimum at (-1, 45)
C) Relative maximum at (-3, -18); relative minimum at (1, 109)
D) Relative maximum at (-3, 109); relative minimum at (1, -19)

Question 49

The second derivative test reveals that f (x) =x²- 5 has

A) a relative maximum at $x = \sqrt { 5 }$
B) a point of inflection at $x = \sqrt { 5 }$
C) a relative minimum at x = 0.
D) nothing significant at x = 5.

Question 50

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Question 51

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 4 x ^ { 2 } + 1$ has

A) a relative maximum only.
B) a relative minimum only.
C) a relative maximum and two relative minima.
D) neither a relative maximum nor a relative minimum.

Question 52

The second derivative test reveals that $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$ has

A) a relative maximum and two relative minima.
B) a relative maximum only.
C) a relative minimum only.
D) neither a relative maximum nor a relative minimum.

Question 53

A manufacturer estimates that if he produces x units of a particular commodity, the total cost will be $C ( x ) = x ^ { 3 } - 24 x ^ { 2 } + 350 x + 400$ dollars. For what value of x does the marginal cost $M ( x ) = C ^ { t } ( x )$ satisfy $M ^ { \prime } ( x ) = 0$ ?

Question 54

Name the vertical and horizontal asymptotes of the given graph.

A) Vertical asymptotes: x = -4, x = 1; horizontal asymptote: y = 2
B) Vertical asymptotes: x = -4, x = 1; horizontal asymptotes: y = 2, y = 0
C) Vertical asymptotes: x = -4, x = 1, x = 0; horizontal asymptote: y = 2
D) Vertical asymptotes: x = -4, x = 1, x = 0; horizontal asymptotes: y = 2, y = 0

Question 55

Find all vertical and horizontal asymptotes of the graph of the given function. $$f ( x ) = \frac { x ^ { 2 } - 2 x } { x ^ { 2 } - 8 x + 15 }$$

A) Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 0
B) Vertical asymptotes: x = 5, x = 3, x = 2; horizontal asymptote: y = 1
C) Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 2
D) Vertical asymptotes: x = 5, x = 3; horizontal asymptote: y = 1

Question 56

Graph $f ( x ) = x ^ { 2 } + 4 x + 5$ .

Question 57

Graph $f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }$ .

Question 58

Graph $f ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - x$ .

Question 59

Sketch the graph of the given function. $$f ( x ) = \frac { x - 5 } { x - 3 }$$

A)

B)

C)

D)

Question 60

Find A and B so that the graph of $f ( x ) = \frac { 29 - A x } { B x + 15 }$ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Question 61

Find the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 12$ on the interval -1 $\le$ x $\le$ 2.

Question 62

Determine the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5$ on the interval -2 $\le$ x $\le$ 1.

Question 63

Find the absolute maximum and minimum of the function $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval 0 $\le$ x $\le$ 3.

Question 64

Find the absolute maximum of the function $f ( x ) = x ^ { 3 }$ on the interval $- \frac { 1 } { 2 } \leq x \leq 1$ .

A) 0
B) $\frac { 1 } { 8 }$
C) -1
D) 1

Question 65

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 3.

A) -1
B) -4
C) 0
D) 3

Question 66

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\le$ x $\le$ 7.

A) -4
B) 0
C) 7
D) -1

Question 67

Find the absolute maximum and absolute minimum of $f ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 8 x + 2$ on the interval -2 $\le$ x $\le$ 5.

Question 68

The absolute minimum of the function $f ( x ) = 8 x ^ { 3 } - 4 x ^ { 2 } + 72 x$ on the interval 0 $\le$ x $\le$ 4 is 0.

Question 69

Find the absolute maximum and minimum of the function $f ( x ) = \frac { 1 } { 6 } \left( x ^ { 3 } - 6 x ^ { 2 } + 9 x + 1 \right)$ on the interval 0 $\le$ x $\le$ 2.

Question 70

Find the absolute maximum of the function $f ( x ) = x ^ { 5 } - x ^ { 4 }$ on the interval -1 $\le$ x $\le$ 1.

A) 0
B) 1
C) -1
D) -2

Question 71

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 8$ on the interval -1 $\le$ x $\le$ 2.

Question 72

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 20$ on the interval -9 $\le$ x $\le$ 4.

Question 73

= The absolute maximum of the function $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 3$ on the interval -1 $\le$ x $\le$ 2 is 11.

Question 74

=The absolute maximum of the function $f ( x ) = \frac { 2 x } { x + 1 }$ on the interval 0 $\le$ x $\le$ 1 is 1.

Question 75

The absolute minimum of $f ( x ) = \frac { 9 } { x } + x - 3$ on the interval 1 $\le$ x $\le$ 9 is 1.

Question 76

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 4 x + 7$ dollars. If the price is p(x) = (49 - x) dollars per unit, determine the level of production that maximizes profit.

A) x = 1
B) x = 2
C) x = 3
D) x = 6

Question 77

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula $C = 0.1 x ^ { 2 } + 20 x + 500$ . Find the number of units that will minimize the average cost.

A) 100
B) 147
C) 36
D) 71

Question 78

Find the elasticity n of the demand function $D ( p ) = \frac { 3 } { 1 + 2 p ^ { 2 } }$ .

A) $n = - \frac { 6 } { 1 + 2 p ^ { 2 } }$
B) $n = - \frac { 4 p ^ { 2 } } { 1 + 2 p ^ { 2 } }$
C) n = 4p
D) $n = - 4 p ^ { 3 }$

Question 79

Suppose the total cost of producing x units of a certain commodity is $C ( x ) = 2 x ^ { 4 } - 10 x ^ { 3 } - 18 x ^ { 2 } + 200 x + 167$ . Determine the largest and smallest values of the marginal cost for 0 $\le$ x $\le$ 5.

Question 80

The total cost of producing x units of a certain commodity is $C ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 8 x$ . Determine the minimum average cost of the commodity.