Deck 5: Applications of the Derivative

Question 1

Let $V = \frac { 4 } { 3 } \pi r ^ { 3 }$ If $\frac { d r } { d t } = 2$ when $r = 1 \text {, }$ then $\frac { d V } { d t }$ is

A) $16 \pi$
B) $12 \pi$
C) $8 \pi$
D) $4 \pi$
E) $2 \pi$

Answer 1

$8 \pi$

Question 2

Let $A = \pi r ^ { 2 }$ If $\frac { d r } { d t } = 1.5$ when then $r = 6,$ is

A) $18 \pi$
B) $16 \pi$
C) $12 \pi$
D) $8 \pi$
E) $4 \pi$

Answer 2

$18 \pi$

Question 3

Let $s ^ { 2 } = 200 + x ^ { 2 }$ If $\frac { d x } { d t } = - 6$ when $x = 5 \text {, }$ then $\frac { d s } { d t }$ is

A)-6
B)-4
C)-2
D)2
E)4

Answer 3

-2

Question 4

Let $x y = 1$ If $\frac { d x } { d t } = 12$ when $x = 2$ and $y = \frac { 1 } { 2 } ,$ then $\frac { d y } { d t }$ is

A)-8
B)-6
C)-4
D)-3
E)-2

Question 5

Let $A = \frac { 1 } { 2 } b h .$ If $\frac { d h } { d t } = - 4 , \frac { d b } { d t } = 6,$ when and $h = 6,$ then $\frac { d A } { d t }$ is

A)12
B)10
C)8
D)6
E)4

Question 6

Let $x ^ { 2 } + y ^ { 2 } = 25$ If $\frac { d y } { d t } = - 4$ when $x = 4$ and $y = 3 \text {, }$ then $\frac { d x } { d t }$ is

A)5
B)3
C) $\frac { 3 } { 2 }$
D)1
E) $\frac { 1 } { 2 }$

Question 7

Let $V = \frac { 3 } { 16 } \pi h ^ { 3 }$ If $\frac { d V } { d t } = - 72$ when $h = 12$ then $\frac { d h } { d t }$ is

A) $\frac { 4 } { 3 \pi }$
B) $\frac { 8 } { 3 \pi }$
C) $- \frac { 8 } { 3 \pi }$
D) $\frac { 8 } { 9 \pi }$
E) $- \frac { 8 } { 9 \pi }$

Question 8

Let $s ^ { 2 } = x ^ { 2 } + 9$ If $\frac { d x } { d t } = 4$ when $x = 4 ,$ then $\frac { d s } { d t }$ is

A) $\frac { 5 } { 16 }$
B) $\frac { 1 } { 5 }$
C)5
D) $\frac { 16 } { 5 }$
E)16

Question 9

Let $A = ( 10 + x ) ^ { 2 }$ If $\frac { d x } { d t } = \frac { 2 } { 15 }$ when then $\frac { d A } { d t }$ is

A)8
B)6
C)4
D)2
E) $\frac { 1 } { 2 }$

Question 10

Let $x \cos y = 5$ If $\frac { d x } { d t } = - 4$ when $y = - \frac { \pi } { 3 } ,$ then $\frac { d y } { d t }$ is

A) $- \frac { 2 } { 15 } \sqrt { 3 }$
B) $- \frac { 2 } { 15 }$
C) $\frac { 2 } { 15 }$
D) $\frac { 2 } { 15 } \sqrt { 3 }$
E) $\frac { 2 } { 5 } \sqrt { 3 }$

Question 11

Let $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and assume that $z > 0$ If $\frac { d x } { d t } = - 25$ and $\frac { d y } { d t } = - \frac { 50 } { 3 }$ when $x = 200$ and $y = 150 \text {, }$ then $\frac { d z } { d t }$ is

A)-40
B)-30
C)30
D)40
E)50

Question 12

Let $\sin ^ { 2 } x + \cos ^ { 2 } y = \frac { 5 } { 4 }$ If $\frac { d y } { d t } = - \frac { \sqrt { 3 } } { 2 }$ when $x = \frac { 2 \pi } { 3 }$ and $y = \frac { 3 \pi } { 4 },$ then $\frac { d x } { d t }$ is

A)-1
B) $- \frac { 1 } { 2 }$
C)1
D) $\frac { 1 } { 2 }$
E) $\frac { 3 } { 2 }$

Question 13

Let $\sqrt { x } + \sqrt { y } = 5$ If $\frac { d x } { d t } = - \frac { 3 } { 4 }$ when $x = 1 ,$ then $\frac { d y } { d t }$ is

A)4
B)3
C)2
D)-2
E)-3

Question 14

Let $2 x + 3 y = 8$ If $\frac { d y } { d t } = 4,$ then $\frac { d x } { d t }$ is

A)-6
B)-4
C)-2
D)2
E)6

Question 15

Let $\frac { y } { x } = 12$ If $\frac { d x } { d t } = - \frac { 1 } { 2 },$ then $\frac { d y } { d t }$ is

A)-6
B)-4
C)-2
D)2
E)6

Question 16

If each edge of a cube is increasing at the rate of 3 cm/s when the length of an edge is 2 cm long, then the rate of increase of its volume is

A) $6 \mathrm {~cm} ^ { 3 } / \mathrm { s }$
B) $12 \mathrm {~cm} ^ { 3 } / \mathrm { s }$
C) $24 \mathrm {~cm} ^ { 3 } / \mathrm { s }$
D) $36 \mathrm {~cm} ^ { 3 } / \mathrm { s }$
E) $72 \mathrm {~cm} ^ { 3 } / \mathrm { s }$

Question 17

If the surface area of a sphere is shrinking at the rate of 0.1 sq cm/h when the radius is $\frac { 20 } { \pi } \mathrm { cm } \text {, }$ the rate of change of the radius in cm/s is

A) $- \frac { 1 } { 400 }$
B) $- \frac { 1 } { 800 }$
C) $- \frac { 1 } { 1600 }$
D) $- \frac { 1 } { 3200 }$
E) $- \frac { 1 } { 6400 }$

Question 18

An isosceles triangle has equal sides 4 cm long and the included angle ?. If $\frac { d \theta } { d t } = 2 \mathrm { rad } / \mathrm { min }$ when $\theta = \frac { \pi } { 3 },$ then the rate of change of the area in cm²/min is

A)-4
B)-8
C)-16
D)-32
E)-64

Question 19

Water is flowing into a vertical cylindrical tank at the rate of 5 m³/min. If the radius of the tank is 3 m, then the rate at which the height of the water is rising is

A) $\frac { 9 } { 5 \pi }$
B) $\frac { 3 } { 5 \pi }$
C) $\frac { 5 } { 6 \pi }$
D) $\frac { 5 } { 9 \pi }$
E) $\frac { 5 } { 3 \pi }$

Question 20

A spherical balloon is inflated at the rate of 10 m³/min when the radius is 3 m. The rate of increase of the surface area of the balloon is

A) $\frac { 80 } { 3 }$
B) $\frac { 40 } { 3 }$
C) $\frac { 20 } { 3 }$
D) $\frac { 20 } { 3 }$ .
E) $\frac { 5 } { 3 }$

Question 21

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ The set of all critical numbers of ƒ is

A) $\left\{ - 1 , - \frac { 1 } { 3 } \right\}$
B) $\left\{ - 1 , \frac { 1 } { 3 } \right\}$
C) $\left\{ 1 , \frac { 1 } { 3 } \right\}$
D) $\left\{ 1 , - \frac { 1 } { 3 } \right\}$
E) $\{ 1 \}$

Question 22

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 36 x + 5$ The set of all critical numbers of ƒ is

A) $\{ - 2 , - 3 \}$
B) $\{ - 2,3 \}$
C) $\{ 2 , - 3 \}$
D) $\{ 2,3 \}$
E) $\{ 1,3 \}$

Question 23

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 1$ The set of all critical numbers of ƒ is

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

Question 24

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 } { 2 } x ^ { 2 } + 1$ The set of all critical numbers of ƒ is

A) $\{ - \sqrt { 3 } , \sqrt { 3 } \}$
B) $\{ - \sqrt { 3 } , 0 \}$
C) $\{ 0 , \sqrt { 3 } \}$
D) $\{ 0 \}$
E) $\{ - \sqrt { 3 } , 0 , \sqrt { 3 } \}$

Question 25

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ The set of all critical numbers of ƒ is

A) $\{ - 2 \}$
B) $\{ 2 \}$
C) $\{ - 2,2 \}$
D) $\{ 0 , - 2 \}$
E) $\{ 0,2 \}$

Question 26

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ The set of all critical numbers of ƒ is

A) $\{ - 2 \}$
B) $\{ 2 \}$
C) $\{ - 2,2 \}$
D) $\{ 0,2 \}$
E) $\{ 0,1 \}$

Question 27

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ The set of all critical numbers of ƒ is

A) $\{ - 2 \}$
B) $\{ 2 \}$
C) $\{ - 2,2 \}$
D) $\{ 0 , - 2 \}$
E) $\varnothing$

Question 28

Let $f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } + 3 x - \frac { 1 } { 5 }$ The set of all critical numbers of ƒ is

A) $\{ - 1 \}$
B) $\{ - 5 \}$
C) $\{ - 1 , - 5 \}$
D) $\{ 1,5 \}$
E) $\varnothing$

Question 29

Let $f ( x ) = \frac { 1 } { 4 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } - 3 x ^ { 2 } + 1$ The set of all critical numbers of ƒ is

A) $\{ - 2,0 \}$
B) $\{ 2,3 \}$
C) $\{ 0 , - 3 \}$
D) $\{ - 2,0,3 \}$
E) $\{ - 2,0 \}$ .

Question 30

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The set of all critical numbers of ƒ is

A) $\{ 0,2 \pi \}$
B) $\left\{ \frac { \pi } { 2 } \right\}$
C) $\{ \pi \}$
D) $\left\{ \frac { 3 \pi } { 2 } \right\}$
E) $\{ 2 \pi \}$

Question 31

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒare located respectively at x =

A)-2,2
B)-1,2
C) $- \frac { 1 } { 3 } , 2$
D) $- 2,1$
E) $- \frac { 1 } { 3 } , 1$

Question 32

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A)1,4
B)1,2
C)1,3
D)2,3
E)3,4

Question 33

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 1$ on $[ - 1,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A) $\frac { 3 } { 2 } , - 1$
B) $- 1 , \frac { 3 } { 2 }$
C) $\frac { 3 } { 2 } , 2$
D) $2 , \frac { 3 } { 2 }$
E)0,2

Question 34

Let $f ( x ) = \frac { x ^ { 4 } } { 4 } - \frac { 3 x ^ { 2 } } { 2 } + 1$ on $[ 0,2 ]$ . The absolute minimum and maximum of ƒ are located respectively at x =

A)0,2
B)0,1
C) $\sqrt { 3 } , 0$
D) $\sqrt { 3 } , 2$
E)1,2

Question 35

Let $f ( x ) = x ^ { 3 } - 12 x + 1$ on $[ - 3,0 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

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

Question 36

Let $f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & - 2 \leq x < 1 \\x ^ { 3 } & 1 \leq x \leq 2\end{array} \right.$ on $[ - 2,2 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A) $- 2,0$
B) $0 , - 2$
C) $0,2$
D) $- 2,2$
E) $- 3 , - 2$

Question 37

Let $f ( x ) = \frac { 1 } { 2 } x ^ { 3 } - 3 x ^ { 2 } + 6 x - 4$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A)1,3
B)3,1
C)1,2
D)2,3
E)3,2

Question 38

Let $f ( x ) = \frac { 1 } { 5 } x ^ { 3 } - \frac { 9 } { 5 } x ^ { 2 } - 3 x - \frac { 1 } { 5 }$ on $[ 1,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A)0,3
B)3,0
C)1,3
D)3,1
E)2,3

Question 39

Let $f ( x ) = \frac { 1 } { 4 } x ^ { 4 } - \frac { 1 } { 3 } x ^ { 3 } - x ^ { 2 } + 1$ on $[ - 2,3 ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A)0,-2
B)0,2
C)1,2
D)2,-2
E)3,0

Question 40

Let $f ( x ) = x - \sin x$ on $[ 0,2 \pi ]$ The absolute minimum and maximum of ƒ are located respectively at x =

A) $0 , \frac { \pi } { 2 }$
B) $0 , \pi$
C) $\frac { 3 \pi } { 2 } , \frac { \pi } { 2 }$
D) $\frac { \pi } { 2 } , 2 \pi$
E) $0,2 \pi$

Question 41

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ' is given below:
On what interval(s) is the graph of ƒ increasing?

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

Question 42

Let y = ƒ(x) be a differentiable function for which the graph of its derivative, ƒ', is given below:
At what x-value(s), if any, does the graph of ƒ have a horizontal tangent?
​

A)x = 0 only
B)x = -3 only
C)x = 3 only
D)x = -3 and x = 3
E)None

Question 43

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of ƒ have a vertical tangent?

A)x = 0 only
B)x = -3 only
C)x = 3 only
D)x = -3 and x = 3
E)None

Question 44

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
On what interval(s) is the graph of f decreasing?

A) $( - \infty , 1 )$
B) $( - \infty , - 2 ) \cup ( 0 , \infty )$
C)(-2, 0)
D) $( 1 , \infty )$
E)(-2, 1)

Question 45

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a horizontal tangent?

A)x = -1 and x = 0
B)x = -2
C)x = -2, x = 1, and x = 0
D)x = -2 and x = 1
E)None

Question 46

Let $f ( x ) = x ^ { 2 } - 3 x \text { on } [ 0,3 ]$ on $[ 0,3 ]$ . Then the set of all c in (0,3) guaranteed by Rolle's Theorem is

A) $\{ - 3 \}$
B) $\left\{ - \frac { 3 } { 2 } \right\}$
C) $\left\{ - \frac { 1 } { 2 } \right\}$
D) $\left\{ \frac { 1 } { 2 } \right\}$
E) $\left\{ \frac { 3 } { 2 } \right\}$

Question 47

Let $f ( x ) = x ^ { 2 } - 2 x - 2$ on $[ 0,2 ]$ . Then the set of all c in (0,2) guaranteed by Rolle's Theorem is

A) $\{ - 2 \}$
B) $\{ - 1 \}$
C) $\{ 0 \}$
D) $\{ 1 \}$
E) $\{ 2 \}$

Question 48

Let $f ( x ) = x ^ { 3 } - 4 x$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

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

Question 49

Let $f ( x ) = x ^ { 3 } - x + 2$ on $[ - 1,1 ]$ Then the set of all c in (-1,1) guaranteed by Rolle's Theorem is

A) $\left\{ \frac { - \sqrt { 3 } } { 3 } \right\}$
B) $\left\{ \frac { \sqrt { 3 } } { 3 } \right\}$
C) $\left\{ \frac { - \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right\}$
D) $\{ - 1,1 \}$
E) $\{ 1 \}$

Question 50

Let $f ( x ) = x ^ { 4 } - 3$ on $[ - 2,2 ]$ Then the set of all c in $( - 2,2 )$ guaranteed by Rolle's Theorem is

A) $\{ - 1 \}$
B) $\left\{ - \frac { \sqrt { 3 } } { 3 } \right\}$
C) $\left\{ \frac { \sqrt { 3 } } { 3 } \right\}$
D) $\{ 0 \}$
E) $\{ 1 \}$

Question 51

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 1$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

A) $\{ - 1 \}$
B) $\{ 0 \}$
C) $\{ 1 \}$
D) $\{ 0,1 \}$
E) $\{ - 1,0,1 \}$

Question 52

Let $f ( x ) = x ^ { 4 } + x ^ { 2 }$ on $[ - 2,2 ]$ Then the set of all c in (-2,2) guaranteed by Rolle's Theorem is

A) $\{ - 1 \}$
B) $\{ 0 \}$
C) $\{ 1 \}$
D) $\{ 0,1 \}$
E) $\{ - 1,0,1 \}$

Question 53

Let $f ( x ) = \sin x + \cos x$ on $[ 0,2 \pi ]$ Then the set of all in c guaranteed by Rolle's Theorem is

A) $\left\{ \frac { 3 \pi } { 4 } \right\}$
B) $\left\{ \frac { 7 \pi } { 4 } \right\}$
C) $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$
D) $\left\{ \frac { \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$
E) $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right\}$

Question 54

Let $f ( x ) = x ^ { 3 } + 1$ on $[ 0,2 ]$ Then the set of all c in (0,2) guaranteed by the Mean Value Theorem is

A) $\left\{ \frac { \sqrt { 3 } } { 4 } \right\}$
B) $\left\{ \frac { 3 } { 4 } \right\}$
C) $\left\{ \frac { 2 \sqrt { 3 } } { 3 } \right\}$
D) $\left\{ \frac { 5 } { 4 } \right\}$
E) $\{ 1 \}$

Question 55

Let $f ( x ) = x + 2 + \frac { 3 } { x - 1 }$ on $[ 2,7 ]$ Then the set of all c in (2,7) guaranteed by the Mean Value Theorem is

A) $\{ 1 - \sqrt { 6 } \}$
B) $\{ 1 + \sqrt { 6 } \}$
C) $\{ 0 \}$
D) $\{ 1 - \sqrt { 6 } , 1 + \sqrt { 6 } \}$
E) $\{ 1 - \sqrt { 3 } , 0,1 + \sqrt { 3 } \}$

Question 56

Let $f ( x ) = \ln x$ on $[ 1 , e ]$ Then the set of all c in (1,e) guaranteed by the Mean Value Theorem is

A) $\{ 1.5 \}$
B) $\{ e - 1 \}$
C) $\{ 2 \}$
D) $\{ 2.5 \}$
E) $\{ 1.5 , e - 1 \}$

Question 57

Let $f ( x ) = ( x - 4 ) ^ { 2 } - 1$ on $[ 3,6 ]$ Then the set of all c in (3,6) guaranteed by the Mean Value Theorem is

A) $\{ 4 \}$
B) $\left\{ \frac { 7 } { 2 } \right\}$
C) $\{ 2 \}$
D) $\left\{ \frac { 9 } { 2 } \right\}$
E) $\{ 5 \}$

Question 58

Let $f ( x ) = x ^ { 2 } + 2 x - 1$ on $[ 0,1 ]$ . Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

A) $\left\{ \frac { 1 } { 4 } \right\}$
B) $\left\{ \frac { 1 } { 3 } \right\}$
C) $\left\{ \frac { 1 } { 2 } \right\}$
D) $\left\{ \frac { 2 } { 3 } \right\}$
E) $\left\{ \frac { 3 } { 4 } \right\}$

Question 59

Let $f ( x ) = x ^ { \frac { 2 } { 3 } }$ on $[ 0,1 ]$ Then the set of all c in (0,1) guaranteed by the Mean Value Theorem is

A) $\left\{ \frac { 2 } { 9 } \right\}$
B) $\left\{ \frac { 1 } { 3 } \right\}$
C) $\left\{ \frac { 4 } { 9 } \right\}$
D) $\left\{ \frac { 8 } { 27 } \right\}$
E) $\left\{ \frac { 11 } { 27 } \right\}$

Question 60

Let $f ( x ) = x + \frac { 1 } { x } f$ on $\left[ \frac { 1 } { 2 } , 2 \right] .$ Then the set of all c in $\left( \frac { 1 } { 2 } , 2 \right)$ guaranteed by the Mean Value Theorem is

A) $\left\{ \frac { 3 } { 4 } \right\}$
B) $\{ 1 \}$
C) $\left\{ \frac { 5 } { 4 } \right\}$
D) $\left\{ \frac { 3 } { 2 } \right\}$
E) $\left\{ \frac { 7 } { 4 } \right\}$

Question 61

Let $f ( x ) = \sqrt { x - 1 }$ on $[ 1,3 ]$ Then the set of all c in (1,3) guaranteed by the Mean Value Theorem is

A) $\left\{ \frac { 5 } { 4 } \right\}$
B) $\left\{ \frac { 9 } { 8 } \right\}$
C) $\left\{ \frac { 3 } { 2 } \right\}$
D) $\{ 2 \}$
E) $\left\{ \frac { 5 } { 2 } \right\}$

Question 62

The largest set on which the function $f ( x ) = x e ^ { x }$ is increasing is

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

Question 63

The largest set on which the function $f ( x ) = 3 \sin x$ on $[ 0,2 \pi ]$ is increasing is

A) $\left( 0 , \frac { \pi } { 2 } \right)$
B) $\left( \frac { \pi } { 2 } , \pi \right)$
C) $\left( \pi , \frac { 3 \pi } { 2 } \right)$
D) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \frac { 3 \pi } { 2 } , 2 \pi \right)$
E) $( 0,2 \pi )$

Question 64

The largest set on which the function $f ( x ) = 5 ^ { - x }$ is decreasing is

A) $( - 2 , \infty )$
B) $( - \infty , 2 )$
C) $( 2 , e )$
D) $( e , \infty )$
E) $( 2 , e ) \cup ( e , \infty )$

Question 65

The largest set on which the function $f ( x ) = 5 ^ { - x }$ is decreasing is

A) $( - \infty , \infty )$
B) $( - \infty , - 5 )$
C) $( - 5 , \infty )$
D) $( - 5,5 )$
E) $( 5 , \infty )$

Question 66

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a local maximum?

A)x = 0 only
B)x = -3 only
C)x = 3 only
D)x = -3 and x = 3
E)None

Question 67

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
On what interval(s) is the graph of f concave up?

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

Question 68

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a local minimum?

A) x = 3 only
B)x = -3 only
C)x =0 only
D)x = -3 and x = 3
E)None

Question 69

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
At what x-value(s), if any, does the graph of f have a local minimum?

A)x = 0
B)x = -2
C)x = -2, x = 1, and x = 0
D)x = 1
E)None

Question 70

Let $y = f ( x )$ be a differentiable function for which the graph of its derivative, ƒ ', is given below:
On what interval(s) is the graph of f concave down?

A) $( - \infty , 1 )$
B) $( - \infty , - 2 ) \cup ( 0 , \infty )$
C)(-2, 0)
D) $( 1 , \infty )$
E)(-2, 1)

Question 71

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ Then ƒ has a relative maximum at x =

A) $- \frac { 1 } { 3 }$
B)1
C) $\frac { 1 } { 3 }$
D)-1
E)2

Question 72

Let $f ( x ) = x ^ { 3 } - x ^ { 2 } - x + 8$ Then ƒ has a relative minimum at x =

A) $- \frac { 1 } { 3 }$
B)1
C) $\frac { 1 } { 3 }$
D)-1
E)2

Question 73

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2$ Then ƒ has a relative maximum at x =

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

Question 74

Let $f ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } - 36 x - 2$ Then ƒ has a relative minimum at x =

A)2
B)4
C)6
D)8
E)10

Question 75

Let $f ( x ) = x ^ { 4 } - 2 x ^ { 3 } + 3$ Then ƒ has a relative minimum at x =

A)0
B)1
C) $\frac { 2 } { 3 }$
D) $\frac { 3 } { 2 }$
E)2

Question 76

Let $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$ Then ƒ has a relative maximum at x =

A)0
B)1
C) $\frac { 2 } { 3 }$
D) $\frac { 3 } { 2 }$
E)2

Question 77

Let $f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 15 x$ Then ƒ has a relative minimum at x =

A)0
B)1
C)2
D)4
E)5

Question 78

Let $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 16 x$ Then ƒ has a relative minimum at x =

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

Question 79

Let $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1$ Then ƒ has a relative maximum at x =

A)-3
B)-2
C)-1
D)0
E)2

Question 80

Let $f ( x ) = - x ^ { 3 } - 6 x ^ { 2 } - 9 x + 1$ Then ƒ has a relative minimum at x =

A)-3
B)-2
C)-1
D)0
E)2