Problem 1

Find the locations of any absolute extrema for the functions with graphs as follows.

GRAPH

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Problem 2

Find the locations of any absolute extrema for the functions with graphs as follows.

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Problem 3

Find the locations of any absolute extrema for the functions with graphs as follows.

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Problem 4

Find the locations of any absolute extrema for the functions with graphs as follows.

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Problem 5

Find the locations of any absolute extrema for the functions with graphs as follows.

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Problem 6

Find the locations of any absolute extrema for the functions with graphs as follows.

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Problem 7

Find the locations of any absolute extrema for the functions with graphs as follows.

GRAPH

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Problem 8

Find the locations of any absolute extrema for the functions with graphs as follows.

GRAPH

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Problem 9

What is the difference between a relative extremum and an absolute extremum?

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Problem 10

Can a relative extremum be an absolute extremum? Is a relative extremum necessarily an absolute extremum?

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Problem 11

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{3}-6 x^{2}+9 x-8 ; [0,5]$$

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Problem 12

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{3}-3 x^{2}-24 x+5 ; [-3,6]$$

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Problem 13

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{1}{3} x^{3}+\frac{3}{2} x^{2}-4 x+1 ; [-4,2]$$

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Problem 14

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}-6 x+3 ; [-4,4]$$

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Problem 15

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{4}-18 x^{2}+1 ; [-4,4]$$

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Problem 16

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{4}-32 x^{2}-7 ; [-5,6]$$

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Problem 17

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{1-x}{3+x} ; [0,3]$$

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Problem 18

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{8+x}{8-x} ;[4,6]$$

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Problem 19

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{x-1}{x^{2}+1} ; [1,5]$$

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Problem 20

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{x}{x^{2}+2} ; [0,4]$$

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Problem 21

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\left(x^{2}-4\right)^{1 / 3} ; [-2,3]$$

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Problem 22

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\left(x^{2}-16\right)^{2 / 3} ; [-5,8]$$

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Problem 23

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=5 x^{2 / 3}+2 x^{5 / 3} ; d[-2,1]$$

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Problem 24

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x+3 x^{2 / 3} ; [-10,1]$$

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Problem 25

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{2}-8 \ln x ;[1,4]$$

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Problem 26

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=\frac{\ln x}{x^{2}} ;[1,4]$$

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Problem 27

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x+e^{-3 x} ;[-1,3]$$

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Problem 28

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.

$$f(x)=x^{2} e^{-0.5 x} ; [2,5]$$

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Problem 29

Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema.

$$f(x)=\frac{-5 x^{4}+2 x^{3}+3 x^{2}+9}{x^{4}-x^{3}+x^{2}+7} ;[-1,1]$$

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Problem 30

Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema.

$$f(x)=\frac{x^{3}+2 x+5}{x^{4}+3 x^{3}+10} ;[-3,0]$$

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Problem 31

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=2 x+\frac{8}{x^{2}}+1, x>0$$

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Problem 32

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=12-x-\frac{9}{x}, x>0$$

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Problem 33

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=-3 x^{4}+8 x^{3}+18 x^{2}+2$$

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Problem 34

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=x^{4}-4 x^{3}+4 x^{2}+1$$

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Problem 35

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=\frac{x-1}{x^{2}+2 x+6}$$

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Problem 36

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=\frac{x}{x^{2}+1}$$

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Problem 37

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=\frac{\ln x}{x^{3}}$$

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Problem 38

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.

$$f(x)=x \ln x$$

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Problem 39

Find the absolute maximum and minimum of $f(x)=$ $2 x-3 x^{2 / 3}(a)$ on the interval $[-1,0.5] ;(b)$ on the interval $[0.5,2] .$

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Problem 40

Let $f(x)=e^{-2 x} .$ For $x>0,$ let $P(x)$ be the perimeter of the rectangle with vertices $(0,0),(x, 0),(x, f(x))$ and $(0, f(x)) .$ Which of the following statements is true? Society of Actuaries.

a. The function $P$ has an absolute minimum but not an absolute maximum on the interval $(0, \infty) .$

b. The function $P$ has an absolute maximum but not an absolute minimum on the interval (0, $\infty$ ).

c. The function $P$ has both an absolute minimum and an absolute maximum on the interval $(0, \infty)$ .

d. The function $P$ has neither an absolute maximum nor an absolute minimum on the interval $(0, \infty),$ but the graph of the function $P$ does have an inflection point with positive $x$ -coordinate.

e. The function $P$ has neither an absolute maximum nor an absolute minimum on the interval $(0, \infty),$ and the graph of the function $P$ does not have an inflection point with positive $x$ -coordinate.

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Problem 41

Bank Robberies The number of bank robberies in the United States for the years 2000–2009 is given in the following figure. Consider the closed interval [2000, 2009]. Source: FBI.

a. Give all relative maxima and minima and when they occur on the interval.

b. Give the absolute maxima and minima and when they occur on the interval. Interpret your results.

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Problem 42

Bank Burglaries The number of bank burglaries (entry into or theft from a bank during nonbusiness hours) in the United States for the years 2000–2009 is given in the figure on the following page. Consider the closed interval [2000, 2009]. Source: FBI.

a. Give all relative maxima and minima and when they occur on the interval.

b. Give the absolute maxima and minima and when they occur on the interval. Interpret your results.

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Problem 43

Profit The total profit $P(x)$ (in thousands of dollars) from the sale of $x$ hundred thousand automobile tires is approximated by

$$P(x)=-x^{3}+9 x^{2}+120 x-400, \quad x \geq 5$$

Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit.

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Problem 44

Profit A company has found that its weekly profit from the sale of $x$ units of an auto part is given by

$$P(x)=-0.02 x^{3}+600 x-20,000. $$

Production bottlenecks limit the number of units that can be made per week to no more than $150,$ while a long-term contract requires that at least 50 units be made each week. Find the maximum possible weekly profit that the firm can make.

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Problem 45

Average Cost Find the minimum value of the average cost for the given cost function on the given intervals.

$C(x)=x^{3}+37 x+250$ on the following intervals.

a. $1 \leq x \leq 10 \qquad$ b. $10 \leq x \leq 20$

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Problem 46

Average Cost Find the minimum value of the average cost for the given cost function on the given intervals.

$C(x)=81 x^{2}+17 x+324$ on the following intervals.

a. $1 \leq x \leq 10 \qquad$ b. $10 \leq x \leq 20$

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Problem 47

Each graph gives the cost as a function of production level. Use the method of graphical optimization to estimate the production level that results in the minimum cost per item produced.

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Problem 48

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Problem 49

Each graph gives the profit as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.

GRAPH

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Problem 50

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Problem 51

A marshy region used for agricultural drainage has become contaminated with selenium. It has been determined that flushing the area with clean water will reduce the selenium for a while, but it will then begin to build up again. A biologist has found that the percent of selenium in the soil $x$ months after the flushing begins is given by

$$f(x)=\frac{x^{2}+36}{2 x}, \quad 1 \leq x \leq 12$$

When will the selenium be reduced to a minimum? What is the minimum percent?

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Problem 52

Salmon Spawning The number of salmon swimming upstream to spawn is approximated by

$$S(x)=-x^{3}+3 x^{2}+360 x+5000, 6 \leq x \leq 20,$$

where $x$ represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of salmon swimming upstream.

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Problem 53

Molars Researchers have determined that the crown length of first molars in fetuses is related to the postconception age of the tooth as

$$L(t)=-0.01 t^{2}+0.788 t-7.048$$

where $L(t)$ is the crown length (in millimeters) of the molar $t$ weeks after conception. Find the maximum length of the molar $t$ of first molars during weeks 22 through $28 .$ Source: American Journal of Physical Anthropology.

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Problem 54

Fungal Growth Because of the time that many people spend indoors, there is a concern about the health risk of being exposed to harmful fungi that thrive in buildings. The risk appears to increase in damp environments. Researchers have discovered that by controlling both the temperature and the relative humidity in a building, the growth of the fungus $A$ . versicolor can be limited. The relationship between temperature and relative humidity, which limits growth, can be described by

$$R(T)=-0.00007 T^{3}+0.0401 T^{2}-1.6572 T+97.086$$

$$15 \leq T \leq 46,$$

where $R(T)$ is the relative humidity (in percent) and $T$ is the temperature (in degrees Celsius). Find the temperature at which the relative humidity is minimized. Source: Applied and Environmental Microbiology.

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Problem 55

Gasoline Mileage From information given in a recent business publication, we constructed the mathematical model

$$M(x)=-\frac{1}{45} x^{2}+2 x-20, 30 \leq x \leq 65$$

to represent the miles per gallon used by a certain car at a speed of $x$ mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur.

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Problem 56

Gasoline Mileage For a certain sports utility vehicle,

$$M(x)=-0.015 x^{2}+1.31 x-7.3,30 \leq x \leq 60$$

represents the miles per gallon obtained at a speed of $x$ mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur.

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Problem 57

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.

Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$

Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$

Where should the cut be made in order to minimize the sum of the areas enclosed by both figures?

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Problem 58

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.

Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$

Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$

Where should the cut be made in order to make the sum of the areas maximum? (Hint: Remember to use the endpoints of a domain when looking for absolute maxima and minima.)

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Problem 59

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.

Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$

Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$

For the solution to Exercise 57, show that the side of the square equals the diameter of the circle, that is, that the circle can be inscribed in the square.*

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Problem 60

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.

Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$

Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$

Information Content Suppose dots and dashes are transmitted over a telegraph line so that dots occur a fraction $p$ of the time (where $0 < p < 1 )$ and dashes occur a fraction $1-p$ of the time. The information content of the telegraph line is given by $I(p),$ where

$$I(p)=-p \ln p-(1-p) \ln (1-p)$$

a. Show that $I^{\prime}(p)=-\ln p+\ln (1-p)$

b. Set $I^{\prime}(p)=0$ and find the value of $p$ that maximizes the information content.

c. How might the result in part b be used?

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