Calculus for Business, Economics, Life Sciences and Social Sciences

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Chapter 5

Graphing and Optimization - all with Video Answers

Educators

Section 1

First Derivative and Graphs

Problem 1

Refer to the following graph of $y=f(x)$ :
Identify the intervals on which $f(x)$ is increasing.

Dwijendra Rao

Dwijendra Rao

Numerade Educator

Problem 2

Refer to the following graph of $y=f(x)$ :
Identify the intervals on which $f(x)$ is decreasing.

Dwijendra Rao

Numerade Educator

Problem 3

Refer to the following graph of $y=f(x)$ :
Identify the intervals on which $f^{\prime}(x)<0$.

Dwijendra Rao

Numerade Educator

Problem 4

Refer to the following graph of $y=f(x)$ :
Identify the intervals on which $f^{\prime}(x)>0$.

Dwijendra Rao

Numerade Educator

Problem 5

Refer to the following graph of $y=f(x)$ :
Identify the $x$ coordinates of the points where $f^{\prime}(x)=0 .$

Dwijendra Rao

Numerade Educator

Problem 6

Refer to the following graph of $y=f(x)$ :
Identify the $x$ coordinates of the points where $f^{\prime}(x)$ does not exist.

Dwijendra Rao

Numerade Educator

Problem 7

Refer to the following graph of $y=f(x)$ :
Identify the $x$ coordinates of the points where $f(x)$ has a local maximum.

Dwijendra Rao

Numerade Educator

Problem 8

Refer to the following graph of $y=f(x)$ :
Identify the $x$ coordinates of the points where $f(x)$ has a local minimum.

Dwijendra Rao

Numerade Educator

Problem 9

Problems, $f(x)$ is continuous on $(-\infty, \infty)$ and has critical values at $x=a, b, c,$ and $d$. Use the sign chart for $f^{\prime}(x)$ to determine whether f has a local maximum, a local minimum, or neither at each critical value.

Lucas Finney

Numerade Educator

Problem 10

Problems, $f(x)$ is continuous on $(-\infty, \infty)$ and has critical values at $x=a, b, c,$ and $d$. Use the sign chart for $f^{\prime}(x)$ to determine whether f has a local maximum, a local minimum, or neither at each critical value.

Lucas Finney

Numerade Educator

Problem 11

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 12

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 13

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 14

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 15

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 16

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 17

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 18

Match the graph of $f$ with one of the sign charts $a$ - $h$ in the figure.

Lucas Finney

Numerade Educator

Problem 19

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=x^{3}-12 x+8
$$

Lucas Finney

Numerade Educator

Problem 20

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=x^{3}-27 x+30
$$

Lucas Finney

Numerade Educator

Problem 21

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=(x+5)^{1 / 3}
$$

Lucas Finney

Numerade Educator

Problem 22

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=(x-9)^{2 / 3}
$$

Lucas Finney

Numerade Educator

Problem 23

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=\frac{6}{x+2}
$$

Lucas Finney

Numerade Educator

Problem 24

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=\frac{5}{x-4}
$$

Lucas Finney

Numerade Educator

Problem 25

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=|x|
$$

Lucas Finney

Numerade Educator

Problem 26

Find $(A) f^{\prime}(x),(B)$ the critical values off $f,$ and (C) the partition numbers for $f^{\prime}$.
$$
f(x)=\mid x+3
$$

Lucas Finney

Numerade Educator

Problem 27

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=2 x^{2}-4 x
$$

Dwijendra Rao

Numerade Educator

Problem 28

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=-3 x^{2}-12 x
$$

Dwijendra Rao

Numerade Educator

Problem 29

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=-2 x^{2}-16 x-25
$$

Dwijendra Rao

Numerade Educator

Problem 30

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=-3 x^{2}+12 x-5
$$

Dwijendra Rao

Numerade Educator

Problem 31

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x^{3}+4 x-5
$$

Dwijendra Rao

Numerade Educator

Problem 32

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=-x^{3}-4 x+8
$$

Dwijendra Rao

Numerade Educator

Problem 33

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=2 x^{3}-3 x^{2}-36 x
$$

Dwijendra Rao

Numerade Educator

Problem 34

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=-2 x^{3}+3 x^{2}+120 x
$$

Dwijendra Rao

Numerade Educator

Problem 35

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=3 x^{4}-4 x^{3}+5
$$

Dwijendra Rao

Numerade Educator

Problem 36

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x^{4}+2 x^{3}+5
$$

Dwijendra Rao

Numerade Educator

Problem 37

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=(x-1) e^{-x}
$$

Dwijendra Rao

Numerade Educator

Problem 38

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x \ln x-x
$$

Dwijendra Rao

Numerade Educator

Problem 39

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=4 x^{1 / 3}-x^{2 / 3}
$$

Dwijendra Rao

Numerade Educator

Problem 40

Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=\left(x^{2}-9\right)^{2 / 3}
$$

Dwijendra Rao

Numerade Educator

Problem 41

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x^{4}-4 x^{3}+9 x
$$

Lucas Finney

Numerade Educator

Problem 42

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x^{4}+5 x^{3}-15 x
$$

Lucas Finney

Numerade Educator

Problem 43

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=x \ln x-(x-2)^{3}
$$

Lucas Finney

Numerade Educator

Problem 44

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=e^{-x}-3 x^{2}
$$

Lucas Finney

Numerade Educator

Problem 45

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=e^{x}-2 x^{2}
$$

Lucas Finney

Numerade Educator

Problem 46

Use a graphing calculator to approximate the critical values of $f(x)$ to two decimal places. Find the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema.
$$
f(x)=\frac{\ln x}{x}-5 x+x^{2}
$$

Lucas Finney

Numerade Educator

Problem 47

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=4+8 x-x^{2}
$$

Dwijendra Rao

Numerade Educator

Problem 48

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=2 x^{2}-8 x+9
$$

Dwijendra Rao

Numerade Educator

Problem 49

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=x^{3}-3 x+1
$$

Dwijendra Rao

Numerade Educator

Problem 50

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=x^{3}-12 x+2
$$

Dwijendra Rao

Numerade Educator

Problem 51

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=10-12 x+6 x^{2}-x^{3}
$$

Dwijendra Rao

Numerade Educator

Problem 52

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=x^{3}+3 x^{2}+3 x
$$

Dwijendra Rao

Numerade Educator

Problem 53

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=x^{4}-18 x^{2}
$$

Dwijendra Rao

Numerade Educator

Problem 54

Find the intervals on which $f(x)$ is increasing and the intervals on which $f(x)$ is decreasing. Then sketch the graph. Add horizontal tangent lines.
$$
f(x)=-x^{4}+50 x^{2}
$$

Dwijendra Rao

Numerade Educator

Problem 55

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.

Lucas Finney

Numerade Educator

Problem 56

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$$
\begin{array}{lrrrrr}
\boldsymbol{x} & -2 & -1 & 0 & 1 & 2 \\
\boldsymbol{f}(\boldsymbol{x}) & 1 & 3 & 2 & 1 & -1
\end{array}
$$

Lucas Finney

Numerade Educator

Problem 57

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$$
\begin{array}{lrrrrr}
\boldsymbol{x} & -2 & -1 & 0 & 2 & 4 \\
\boldsymbol{f}(\boldsymbol{x}) & 2 & 1 & 2 & 1 & 0
\end{array}
$$

Lucas Finney

Numerade Educator

Problem 58

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$$
\begin{array}{lrrrrr}
\boldsymbol{x} & -2 & -1 & 0 & 2 & 3 \\
\boldsymbol{f}(\boldsymbol{x}) & -3 & 0 & 2 & -1 & 0
\end{array}
$$

Lucas Finney

Numerade Educator

Problem 59

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$f(-2)=4, f(0)=0, f(2)=-4$
$f^{\prime}(-2)=0, f^{\prime}(0)=0, f^{\prime}(2)=0$
$f^{\prime}(x)>0$ on $(-\infty,-2)$ and $(2, \infty)$
$f^{\prime}(x)<0$ on (-2,0) and (0,2)

Lucas Finney

Numerade Educator

Problem 60

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$f(-2)=-1, f(0)=0, f(2)=1$
$f^{\prime}(-2)=0, f^{\prime}(2)=0$
$f^{\prime}(x)>0$ on $(-\infty,-2),(-2,2),$ and $(2, \infty)$

Lucas Finney

Numerade Educator

Problem 61

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$f(-1)=2, f(0)=0, f(1)=-2$
$f^{\prime}(-1)=0, f^{\prime}(1)=0, f^{\prime}(0)$ is not defined;
$f^{\prime}(x)>0$ on $(-\infty,-1)$ and $(1, \infty)$
$f^{\prime}(x)<0$ on (-1,0) and (0,1)

Lucas Finney

Numerade Educator

Problem 62

Problems, $f(x)$ is continuous on $(-\infty, \infty) .$ Use the given in formation to sketch the graph of $f$.
$f(-1)=2, f(0)=0, f(1)=2$
$f^{\prime}(-1)=0, f^{\prime}(1)=0, f^{\prime}(0)$ is not defined;
$f^{\prime}(x)>0$ on $(-\infty,-1)$ and (0,1)
$f^{\prime}(x)<0$ on (-1,0) and $(1, \infty)$

Lucas Finney

Numerade Educator

Problem 63

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{1}
$$

Aman Gupta

Numerade Educator

Problem 64

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{2}
$$

Aman Gupta

Numerade Educator

Problem 65

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{3}
$$

Aman Gupta

Numerade Educator

Problem 66

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{4}
$$

Aman Gupta

Numerade Educator

Problem 67

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{5}
$$

Aman Gupta

Numerade Educator

Problem 68

Involve functions $f_{1}-f_{6}$ and their derivatives, $g_{1}-g_{6} .$ Use the graphs shown in figures $(A)$ and $(B)$ to match each function $f_{i}$ with its derivative $g_{j}$.
$$
f_{6}
$$

Aman Gupta

Numerade Educator

Problem 69

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 70

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 71

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 72

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 73

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 74

Use the given graph of $y=f^{\prime}(x)$ to find the intervals on which fis increasing, the intervals on which $f$ is decreasing, and the local extrema. Sketch a possible graph of $y=f(x)$.

Lucas Finney

Numerade Educator

Problem 75

Use the given graph of $y=f(x)$ to find the intervals on which $f^{\prime}(x)>0,$ the intervals on which $f^{\prime}(x)<0$ and the values of $x$ for which $f^{\prime}(x)=0 .$ Sketch a possible graph of $y=f^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 76

Use the given graph of $y=f(x)$ to find the intervals on which $f^{\prime}(x)>0,$ the intervals on which $f^{\prime}(x)<0$ and the values of $x$ for which $f^{\prime}(x)=0 .$ Sketch a possible graph of $y=f^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 77

Use the given graph of $y=f(x)$ to find the intervals on which $f^{\prime}(x)>0,$ the intervals on which $f^{\prime}(x)<0$ and the values of $x$ for which $f^{\prime}(x)=0 .$ Sketch a possible graph of $y=f^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 78

Use the given graph of $y=f(x)$ to find the intervals on which $f^{\prime}(x)>0,$ the intervals on which $f^{\prime}(x)<0$ and the values of $x$ for which $f^{\prime}(x)=0 .$ Sketch a possible graph of $y=f^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 79

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=x+\frac{4}{x}
$$

Lucas Finney

Numerade Educator

Problem 80

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=\frac{9}{x}+x
$$

Lucas Finney

Numerade Educator

Problem 81

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=1+\frac{1}{x}+\frac{1}{x^{2}}
$$

Lucas Finney

Numerade Educator

Problem 82

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=3-\frac{4}{x}-\frac{2}{x^{2}}
$$

Lucas Finney

Numerade Educator

Problem 83

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=\frac{x^{2}}{x-2}
$$

Lucas Finney

Numerade Educator

Problem 84

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=\frac{x^{2}}{x+1}
$$

Lucas Finney

Numerade Educator

Problem 85

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=x^{4}(x-6)^{2}
$$

Lucas Finney

Numerade Educator

Problem 86

Find the critical values, the intervals on which $f(x)$ is increasing, the intervals on which $f(x)$ is decreasing, and the local extrema. Do not graph.
$$
f(x)=x^{3}(x-5)^{2}
$$

Lucas Finney

Numerade Educator

Problem 87

Let $f(x)=x^{3}+k x,$ where $k$ is a constant. Discuss the number of local extrema and the shape of the graph of $f$ if
(A) $k>0$
(B) $k<0$
(C) $k=0$

Lucas Finney

Numerade Educator

Problem 88

Let $f(x)=x^{4}+k x^{2},$ where $k$ is a constant. Discuss the number of local extrema and the shape of the graph of $f$ if
(A) $k>0$
(B) $k<0$
(C) $k=0$

Lucas Finney

Numerade Educator

Problem 89

The graph of the total profit $P(x)$ (in dollars) from the sale of $x$ cordless electric screwdrivers is shown in the figure.
(A) Write a brief description of the graph of the marginal profit function $y=P^{\prime}(x)$, including a discussion of any $x$ intercepts.
(B) Sketch a possible graph of $y=P^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 90

The graph of the total revenue $R(x)$ (in dollars) from the sale of $x$ cordless electric screwdrivers is shown in the figure.
(A) Write a brief description of the graph of the marginal revenue function $y=R^{\prime}(x),$ including a discussion of any $x$ intercepts.
(B) Sketch a possible graph of $y=R^{\prime}(x)$.

Aman Gupta

Numerade Educator

Problem 91

The figure approximates the rate of change of the price of bacon over a 70 -month period, where $B(t)$ is the price of a pound of sliced bacon (in dollars) and $t$ is time (in months).
(A) Write a brief description of the graph of $y=B(t)$, including a discussion of any local extrema.
(B) Sketch a possible graph of $y=B(t)$.

Aman Gupta

Numerade Educator

Problem 92

The figure approximates the rate of change of the price of eggs over a 70 -month period, where $E(t)$ is the price of a dozen eggs (in dollars) and $t$ is time (in months).
(A) Write a brief description of the graph of $y=E(t),$ including a discussion of any local extrema.
(B) Sketch a possible graph of $y=E(t)$

Aman Gupta

Numerade Educator

Problem 93

A manufacturer incurs the following costs in producing $x$ water ski vests in one day, for $0<x<150$ : fixed costs, $\$ 320$; unit production cost, $\$ 20$ per vest; equipment maintenance and repairs, $0.05 x^{2}$ dollars So, the cost of manufacturing $x$ vests in one day is given by $$ C(x)=0.05 x^{2}+20 x+320 \quad 0<x<150 $$
(A) What is the average cost $\bar{C}(x)$ per vest if $x$ vests are produced in one day?
(B) Find the critical values of $\bar{C}(x),$ the intervals on which the average cost per vest is decreasing, the intervals on which the average cost per vest is increasing, and the local extrema. Do not graph.

Carson Merrill

Numerade Educator

Problem 94

A manufacturer incurs the following costs in producing $x$ rain jackets in one day for $0<x<200$ : fixed costs, $\$ 450$; unit production cost, $\$ 30$ per jacket; equipment maintenance and repairs, $0.08 x^{2}$ dollars
(A) What is the average cost $\bar{C}(x)$ per jacket if $x$ jackets are produced in one day?
(B) Find the critical values of $\bar{C}(x),$ the intervals on which the average cost per jacket is decreasing, the intervals on which the average cost per jacket is increasing, and the local extrema. Do not graph.

Carson Merrill

Numerade Educator

Problem 95

Show that profit will be increasing over production intervals $(a, b)$ for which marginal revenue is greater than marginal cost.

Lucas Finney

Numerade Educator

Problem 96

Show that profit will be decreasing over production intervals $(a, b)$ for which marginal revenue is less than marginal cost.

Lucas Finney

Numerade Educator

Problem 97

A drug is injected into the bloodstream of a patient through the right arm. The drug concentration in the bloodstream of the left arm $t$ hours after the injection is approximated by $$ C(t)=\frac{0.28 t}{t^{2}+4} \quad 0<t<24 $$. Find the critical values of $C(t),$ the intervals on which the drug concentration is increasing, the intervals on which the concentration of the drug is decreasing, and the local extrema. Do not graph.

Carson Merrill

Numerade Educator

Problem 98

The concentration $C(t),$ in milligrams per cubic centimeter, of a particular drug in a patient's bloodstream is given by $$ C(t)=\frac{0.3 t}{t^{2}+6 t+9} \quad 0<t<12 $$.
where $t$ is the number of hours after the drug is taken orally. Find the critical values of $C(t),$ the intervals on which the drug concentration is increasing, the intervals on which the drug concentration is decreasing, and the local extrema. Do not graph.

Carson Merrill

Numerade Educator