Deck 2: Applications of the Derivative

Question 1

Points A, B, and C lie on the graph of a function f(x), as shown on the diagram. What are the signs of f(x), f'(x), and f''(x) at the point C? Enter your answer as just "neg", "pos", or 0 in the order given above separated by commas.

Answer 1

pos, 0, neg

Question 2

Let F(x) be the function graphed below. For what value is F'(x) > 0?

A) p < x < q
B) 0 < x < q
C) x > p
D) x < 0 and x > p
E) none of these

Answer 2

x > p

Question 3

Is this the graph of a function having the following properties?
(I) asymptotic to the line y = $\frac { 1 } { 2 }$ x - 1
(II) relative maximum at x = 0

Answer 3

True

Question 4

At which labeled point(s) is the graph concave down?

A) B
B) A, B
C) A, B, F
D) A, F

Question 5

Is this the graph of a function having the following properties?
(I) concave down for all x
(II) asymptotic to the line x = 0

Question 6

Which labeled point has the most negative slope?

A) E
B) D
C) F
D) C

Question 7

At which labeled point(s) is the function increasing?

A) A
B) A, E
C) C, F
D) E

Question 8

Which of the following graphs could represent a function with the following properties?
I. f(x) > 0, for x < 0
II. $f ^ { \prime }$ (x) ? 0, for all x
III. $f ^ { \prime }$ (0) = 0

A)
B)
C)
D) none of these

Question 9

Let a, b, and c be fixed numbers with a > 0 and let f(x) = $a x ^ { 2 }$ + bx + c. Which of the following properties is true of the graph of f(x) ?

A) f(x) is always concave up
B) f(x) has either a relative maximum or inflection point
C) f(x) has one relative maximum
D) f(x) has one inflection point
E) none of these

Question 10

Which labeled point has the most positive slope?

A) A
B) E
C) B
D) F

Question 11

Is this the graph of the function having the following properties?
(I) inflection point at x = 3
(II) no relative maximum point
(III) asymptote at x = 5

Question 12

Is this the graph of a function having the following properties?
(I) f'(x) < 0 for all x
(II) f''(x) > 0 for all x < 0
(III) (0, 1) is a point on the graph

Question 13

Is this the graph of a function having the following properties?
(I) defined for x ≥ -1
(II) horizontal asymptote at y = 3
(III) increasing for all x ≥ -1

Question 14

Is this the graph of a function having the following properties?
(I) f'(x) > 0 for all x
(II) f''(x) > 0 for all x < 0, f''(x) < 0 for x >0
(III) asymptotes at y = $\frac { \pi } { 2 }$ , y = - $\frac { \pi } { 2 }$

Question 15

Is this the graph of a function having the following properties?
(I) x-intercept at x = -2
(II) absolute maximum at x = -1
(III) relative maximum at x = 1
(IV) concave up for x ≥ 2

Question 16

At which labeled point(s) is the function decreasing?

A) F
B) A, E
C) C, F
D) C

Question 17

At which labeled point(s) is the graph concave up?

A) D
B) C, E
C) C, D, E
D) C, D

Question 18

Which of the following could represent a function having the given properties?
(I) increasing slope for x < 4
(II) f'(x) > 0 for all x (x ≠ 4)
(III) asymptote at x = 4

A)
B)
C)
D)

Question 19

Is this the graph of a function having the following properties ?
(I) f(0) = -5
(II) f'(0) = 0
(III) f''(0) = 12
(IV) f( $\sqrt { 7 }$ ) = f(- $\sqrt { 7 }$ ) = -12

Question 20

Which of the following graphs could represent a function having the given properties?
(I) asymptotes x = 2 and x = -2
(II) f''(x) > 0 for all x
(III) f(x) > 0 for all x > 4

A)
B)
C)
D)

Question 21

Determine the inflection point of Enter your answer exactly as just an ordered pair of fractions:

Question 22

Find the inflection point(s) of $f ( x ) = \frac { x ^ { 3 } } { 3 } - \frac { 5 } { 2 } x ^ { 2 } + 6 x - 36$

A) $\left( \frac { 5 } { 2 } , \mathrm { f } \left( \frac { 5 } { 2 } \right) \right.$
B) (2, f(2)) and (3, f(3))
C) x = -2, -3
D) (10, f (10))
E) $\left( \frac { 2 } { 5 } , \mathrm { f } \left( \frac { 2 } { 5 } \right) \right.$

Question 23

Find the x coordinates of all relative extreme points of $f ( x ) = \frac { 1 } { 2 } x ^ { 4 } - \frac { 2 } { 3 } x ^ { 3 } - 6 x ^ { 2 } - 100$

A) x = -2, 0, 3
B) x = 0
C) x = -3, 0, 2
D) x = -3, 2
E) x = -2, 3

Question 24

If a function has second derivative f ''(x) = $x ^ { 2 }$ ( $x ^ { 2 }$ - 4) is (0, 0) an inflection point?

Question 25

Find the x coordinates of all relative extreme points of $f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 72$

A) x = 2, 6
B) x = -4, -3, 0
C) x = 3, 4
D) x = 0, 3, 4
E) x = -4, -3

Question 26

Find the inflection point(s) of y = 2 $x ^ { 3 }$ - 3 $x ^ { 2 }$ - 12x + 17.

A) $\left( \frac { 1 } { 2 } , \frac { 21 } { 2 } \right)$
B) $\left( \frac { 1 } { 2 } , \frac { 21 } { 2 } \right)$ and $- \frac { 1 } { 2 } , 22$
C) $\left( \frac { 7 } { 2 } , 24 \right.$ and $\left( \frac { 1 } { 2 } , \frac { 21 } { 2 } \right)$
D) (2, -3)
E) none of these

Question 27

Find the maximum value of the function f(x) = - + 6 + 10 for x ≥ 0.
Enter your answer exactly as: f(a) = b rel max

Question 28

Find the inflection point(s) of $f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 16$ .

A) $\left( \frac { 7 } { 2 } , \mathrm { f } \left( \frac { 7 } { 2 } \right) \right.$
B) (2, f (2)) and (6, f (6))
C) (56, f (56))
D) (3, f (3)) and (4, f (4))
E) $\left( \frac { 2 } { 7 } , \mathrm { f } \left( \frac { 2 } { 7 } \right) \right.$

Question 29

Where is the function f(x) = $\frac { 5 } { ( 2 x - 4 ) ^ { 3 } }$ increasing?

A) for x > 2
B) at x = 30
C) for all x
D) none of these

Question 30

Find the relative minimum point(s) of $f ( x ) = \frac { x ^ { 4 } } { 4 } - x ^ { 3 } - 5 x ^ { 2 } - 10$

A) (-2, f(-2)) and (0, f(0))
B) (2, f(2)) and (-5, f(-5))
C) (-2, f(-2)) and (5, f(5))
D) (0, f(0))
E) none of these

Question 31

Which of the following is (are) true of $f ( x ) = 5 + 3 x ^ { 2 } - x ^ { 3 } ?$
(I) (1, 7) is a point of inflection
(II) f(2) is a relative maximum point
(III) f has a relative minimum point at x = 0
(IV) f is increasing on (2, ?)

A) II, III, and IV
B) I, II, and III
C) II and III
D) all of these
E) none of these

Question 32

Determine the relative maximum point of Enter your answer as exactly just an ordered pair of integers: (a, b)

Question 33

Determine all the values of x where relative maximum and minimum points of the function occur. Distinguish the maxima from the minima using the second derivative rule. Enter your answer exactly as: f(a) rel max, f(b) rel min in that order.

Question 34

Find the relative extreme points for $f ( x ) = x ^ { 3 } + 6 x ^ { 2 } - 15 x$

A) (5, f(5)) is a relative extreme minimum point, (-1, f(-1)) a relative extreme maximum
B) (0, f(0)) is a relative extreme minimum point
C) (5, f(5)) is a relative extreme maximum point, (-1, f(-1)) a relative extreme minimum
D) (-5, f(-5)) is a relative extreme maximum point, (1, f(1)) a relative extreme minimum
E) (-5, f(-5)) is a relative extreme minimum point, (1, f(1)) a relative extreme maximum

Question 35

This is the graph of f(x) = - - 6x. What is the inflection point?
Enter your answer exactly as just an ordered pair of reduced fractions of form , or integers (a, b), or the word "none".

Question 36

Find the interval(s) where f is concave down for $f ( x ) = - 4 x ^ { 4 } + 3 x ^ { 3 } - 5 x + 10$

A) $\left( - \infty , \frac { 3 } { 8 } \right)$
B) (-?, 0) ? (3, ?)
C) $(\left. 0 , \frac { 3 } { 8 } \right)$
D) (-?, 0) ? $\left( \frac { 3 } { 8 } , \infty \right)$
E) (-?, 0)

Question 37

If the second derivative of a function is $f ^ { \prime \prime } ( x ) = ( x - 2 ) ^ { 2 } \left( x ^ { 2 } - 9 \right)$ is (2, f(2)) an inflection point'

Question 38

Find the x coordinates of all relative extreme points of $f ( x ) = \frac { 2 } { 3 } x ^ { 3 } - 7 x ^ { 2 } + 24 x - 72$

A) x = -4, -3, 0
B) x = -4, -3
C) x = 3, 4
D) x = 2, 6
E) x = 0, 3, 4

Question 39

Find the x coordinates of all relative extreme points of $f ( x ) = \frac { 1 } { 4 } x ^ { 4 } + \frac { 2 } { 3 } x ^ { 3 } - \frac { 3 } { 2 } x ^ { 2 } + 4$

A) x = -3, 0, 1
B) x = -1, 0, 3
C) x = -3, 1
D) x = 0
E) x = -1, 3

Question 40

Find the interval(s) where f is concave up for $f ( x ) = 4 x ^ { 4 } - 3 x ^ { 3 } + 5 x - 10$

A) (-?, ?)
B) (-?, 0) ? $\left( \frac { 3 } { 8 } , \infty \right)$
C) $\left( \frac { 3 } { 8 } , \infty \right)$
D) $\left( - \infty , \frac { 3 } { 8 } \right)$
E) $(\left. 0 , \frac { 3 } { 8 } \right)$

Question 41

Which of the following is (are) true of $f ( x ) = 3 x ^ { 2 / 3 } - 2 x ?$
(I) f is decreasing on (1, ?)
(II) (1, f(1)) is a relative minimum point
(III) f is concave down everywhere
(IV) (0, 0) is an inflection point

A) I and III
B) I, II, and III
C) II, III, and IV
D) I, III, and IV
E) all of these

Question 42

Which of the following is (are) true of $f ( x ) = x ^ { 4 } - x ^ { 3 } ?$
(I) (0, 0) is an inflection point
(II) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 16 } \right)$ is an inflection point
(III) (0, f(0)) is a relative minimum point
(IV) f is increasing on (0, 2)

A) I, II, and III
B) I, II, and IV
C) I and II
D) I, III, and IV
E) all of these

Question 43

Which of the following best represents the graph of y = 2 $x ^ { 3 }$ - 3 $x ^ { 2 }$ - 12x + 17 near the point (2, -3)?

A)
B)
C)
D)

Question 44

Is this the graph of f(x) = 9x + 1 + $\frac { 1 } { x }$ , x > 0?

Question 45

Which of the following is the graph of F(x) = 2x - $\frac { 1 } { x + 2 }$ + 5?

A)
B)
C)
D)

Question 46

Is this the graph of the function f(x) = $x ^ { 3 }$ + 3 $x ^ { 2 }$ - 45x + 10?

Question 47

Is this the graph of f(x) = $( x - 2 ) ^ { 3 }$ + 1?

Question 48

Which of the following most resembles the graph of f(x) = $\frac { 1 } { ( 2 - 3 x ) ^ { 3 } }$ near x = $\frac { 2 } { 3 }$ ?

A)
B)
C)
D)

Question 49

Is this the graph of y = $\sqrt { 3 x + 4 }$ ?

Question 50

Which of the following is the graph of f(x) = $x ^ { 4 }$ - 9?

A)
B)
C)
D)

Question 51

Is this the graph of the function f(x) = $- x ^ { 3 }$ + 3 $x ^ { 2 }$ + 9x - 15?

Question 52

Is this the graph of f(x) = $\frac { 1 } { 4 }$ $x ^ { 4 }$ - $x ^ { 3 }$ + 2?

Question 53

Which of the following is (are) true of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 3 x ?$
(I) f increasing on (1, ?)
(II) (1, 1) is a relative extreme point
(III) (1, 1) is an inflection point
(IV) f is concave up on (-?, 1)

A) II, III, and IV
B) I and III
C) I, II, and IV
D) I, II, and III
E) all of these

Question 54

Is this the graph of g(x) = 2 $x ^ { 3 }$ + 3 $x ^ { 2 }$ - 12x - 5?

Question 55

Find the interval(s) where f is increasing and concave down for Enter your answer in standard interval notation as either: (a, b) or (a, b)∪(c, d) where a< c with any irrational numbers in lowest terms of form: e ±

Question 56

Which of the following is the graph of y = $\frac { 1 } { x }$ + 2x + 5?

A)
B)
C)
D)

Question 57

Is this the graph of h(x) = x + 10 + $\frac { 9 } { x }$ ?
Enter just the word "yes" or "no" (lower case).

Question 58

Is this the graph of f(x) = $\frac { 1 } { x + 2 }$ ?

Question 59

Is this the graph of f(x) = $x ^ { 3 }$ - 3 $x ^ { 2 }$ + 2?

Question 60

What are the x-intercepts of the graph of y = 3 $x ^ { 3 }$ + 24?

A) (24, 0)
B) (2, 0)
C) (0, -2)
D) (0, 24)
E) none of these

Question 61

The graph of a function f(x) is sketched below. Which of the following is f(x)?

A) f(x) = $\frac { 1 } { x - 5 }$
B) f(x) = $\sqrt { x - 5 }$ - 2
C) f(x) = $\frac { 1 } { 5 - x }$ + 2
D) f(x) = $x ^ { 2 }$ - 5x - 2

Question 62

Determine the minimum value of x + y when xy = 100 (x > 0, y > 0).
Enter just an integer.

Question 63

A rectangular garden is to be fenced in and divided into three parallel sections. The fencing for the boundary costs $20 per foot whereas the fencing for the dividing fences costs $5 per foot. Consider the problem of finding the dimensions of the largest garden possible if the gardener can spend $2000 for the fencing. Find the values of l and w for which the total area of the garden is as large as possible.

Question 64

A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing $2 per running foot and on one side by a brick wall costing $6 per running foot. Let x be the length of the brick wall side. Which of the following represents the total cost of the material?

A) 3x + (50 - x)
B) 6x + $\frac { 50 } { x }$
C) 6x + $\frac { 300 } { x }$
D) $\frac { 3 x } { 50 }+ \frac { 1 } { 50 x }$
E) none of these

Question 65

Determine the minimum value of f(x) =
Enter just an integer.

Question 66

A homebuilder's advertisement promises a house with a finished recreation room of 300 square feet. Two perpendicular walls of the room are to be paneled at a cost of $5 per running foot. A third side will be built out of windows at a cost of $10 per running foot. What dimensions should the room have to minimize the homebuilder's cost? Enter your answer as: length of side using cinder, length of other side.

Question 67

Determine the minimum value of
Enter just an integer.

Question 68

Suppose f(x) is the function graphed below. Which of the following is f(x)?

A) f(x) = $x ^ { 3 }$ + 5
B) f(x) = $\frac { 1 } { x }$ + $x ^ { 2 }$ + 3x
C) f(x) = $x ^ { 2 }$ + 3x
D) f(x) = $x ^ { 3 }$ + 6 $x ^ { 2 }$ + 9x
E) f(x) = - $x ^ { 2 }$ + 2x + 5

Question 69

A manufacturer estimates that the profit from producing x units of a commodity is $- x ^ { 2 } + 40 x - 100$ dollars per week. What is the maximum profit he can realize in one week?

A) $300
B) $275
C) $500
D) $400
E) none of these

Question 70

Determine the maximum value of f(x) = -2 + 6x + 3 on 0 ≤ x ≤ 2.
Enter just an integer.

Question 71

The graph of a function f(x) is sketched below. Which of the following is f(x)?

A) f(x) = $\frac { 1 } { x }$ - $x ^ { 2 }$
B) f(x) = $\frac { 1 } { x + 2 }$ - x
C) f(x) = - $x ^ { 2 }$ - $x ^ { 3 }$
D) f(x) = $\frac { 1 } { x ^ { 2 } }$

Question 72

Suppose a ball is thrown into the air and after t seconds has a height of $h ( t ) = - 16 t ^ { 2 } + 80 t$ feet. When will it reach its maximum height?

A) 5 sec
B) 300 sec
C) 0.5 sec
D) 2.5 sec
E) none of these

Question 73

A large rectangular garden is to be enclosed by a fence and divided into 5 regions by 4 parallel fences across the interior of the garden as shown below. Find the values of l and w for which the total area of the garden is as large as possible, assuming that 120 ft of fencing is available.

Question 74

Determine all maximum and minimum values of f(x) = - + 3 + 9x - 1 on Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the maximum of f.

Question 75

What is the maximum area that can be enclosed in a rectangular shape with 100 feet of fence if one of the two long sides is not fenced (there is a natural boundary there)?

A) 1000 square feet
B) 2500 square feet
C) 625 square feet
D) 1250 square feet
E) none of these

Question 76

Determine the minimum value of f(x) = - 3 + 2 on 1 ≤ x ≤ 3.
Enter just an integer.

Question 77

Compute the maximum product for two positive numbers with the property that the sum of the first plus five times the second is 5000.

A) 25,000
B) 1,250,000
C) 600
D) 50,000
E) none of these

Question 78

A rectangular corral with a total area of 60 square meters is to be fenced off and then divided into 2 rectangular sections by a fence down the middle. The fencing for the outside costs $9 per running meter, whereas that for the interior dividing fence costs $12 per running meter. Which of the following statements hold, if the cost (c) of the fencing is to be maximized? (I) The constraint equation is 3w + 2 l = 60 .
(II) The objective equation is 2l ∙ w = 60 .
(III) The constraint equation is w ∙ l = 60 .
(IV) The objective equation is C = 30w + 18l .
(V) The constraint equation is C = 12w + 9wl .
(VI) The objective equation is C = 60 - lw .

A) III and IV
B) V and VI
C) I and II
D) none of these

Question 79

Determine the maximum and minimum values of f(x) = 1 - + 3x + 2 on 0 ≤ x ≤ 3 .
Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the maximum of f.

Question 80

Determine the maximum and minimum values of f(x) = - 3 - 9x on -4 ≤ x ≤ 4 .
Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the maximum of f.