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Chapter 4

APPLICATIONS OF DIFFERENTIATION

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Problem 1

Explain the difference between an absolute minimum and a
local minimum.

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Problem 2

Suppose $f$ is a continuous function defined on a closed
interval $[a, b] .$
(a) What theorem guarantees the existence of an absolute
maximum value and an absolute minimum value for $f ?$
(b) What steps would you take to find those maximum and
minimum values?

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Problem 3

$3-4$ . For each of the numbers $a, b, c, d, r,$ and $s$ , state whether
the function whose graph is shown has an absolute maximum or
minimum, a local maximum or minimum, or neither a maximum
nor a minimum.

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Problem 4

$3-4$ . For each of the numbers $a, b, c, d, r,$ and $s$ , state whether
the function whose graph is shown has an absolute maximum or
minimum, a local maximum or minimum, or neither a maximum
nor a minimum.

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Problem 5

$5-6=$ Use the graph to state the absolute and local maximum
and minimum values of the function.

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Problem 6

$5-6=$ Use the graph to state the absolute and local maximum
and minimum values of the function.

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Problem 7

$7-10=$ Sketch the graph of a function $f$ that is continuous
on $[1,5]$ and has the given properties.
Absolute minimum at $2,$ absolute maximum at 3
local minimum at 4

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Problem 8

$7-10=$ Sketch the graph of a function $f$ that is continuous
on $[1,5]$ and has the given properties.
Absolute minimum at $1,$ absolute maximum at $5,$
local maximum at $2,$ local minimum at 4

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Problem 9

$7-10=$ Sketch the graph of a function $f$ that is continuous
on $[1,5]$ and has the given properties.
Absolute maximum at $5,$ absolute minimum at $2,$
local maximum at $3,$ local minima at 2 and 4

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Problem 10

$7-10=$ Sketch the graph of a function $f$ that is continuous
on $[1,5]$ and has the given properties.
$f$ has no local maximum or minimum, but 2 and 4 are
critical numbers

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Problem 11

(a) Sketch the graph of a function that has a local maximum
at 2 and is differentiable at 2 .
(b) Sketch the graph of a function that has a local maximum
at 2 and is continuous but not differentiable at 2 .
(c) Sketch the graph of a function that has a local maximum
at 2 and is not continuous at 2 .

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Problem 12

(a) Sketch the graph of a function on $[-1,2]$ that has an
absolute maximum but no local maximum.
(b) Sketch the graph of a function on $[-1,2]$ that has a
local maximum but no absolute maximum.

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Problem 13

(a) Sketch the graph of a function on $[-1,2]$ that has an
absolute maximum but no absolute minimum.
(b) Sketch the graph of a function on $[-1,2]$ that is discontinuous but has both an absolute maximum and an
absolute minimum.

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Problem 14

(a) Sketch the graph of a function that has two local
maxima, one local minimum, and no absolute minimum.
(b) Sketch the graph of a function that has three local
minima, two local maxima, and seven critical numbers.

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Problem 15

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$$

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Problem 16

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$$

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Problem 17

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=\sin x, \quad 0 \leqslant x<\pi / 2$$

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Problem 18

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

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Problem 19

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=\ln x, \quad 0<x \leqslant 2$$

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Problem 20

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$

$$f(x)=1 / x, \quad 1 < x < 3$$

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Problem 21

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=1-\sqrt{x}$$

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Problem 22

$15-22=$ Sketch the graph of $f$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections $1.2 . )$
$$f(x)=\left\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leqslant x<0} \\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$$

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Problem 23

$23-36=$ Find the critical numbers of the function.
$$f(x)=4+\frac{1}{3} x-\frac{1}{2} x^{2}$$

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Problem 24

$23-36=$ Find the critical numbers of the function.
$$f(x)=x^{3}+6 x^{2}-15 x$$

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Problem 25

$23-36=$ Find the critical numbers of the function.
$$f(x)=2 x^{3}-3 x^{2}-36 x$$

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Problem 26

$23-36=$ Find the critical numbers of the function.
$$f(x)=2 x^{3}+x^{2}+2 x$$

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Problem 27

$23-36=$ Find the critical numbers of the function.
$$g(t)=t^{4}+t^{3}+t^{2}+1$$

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Problem 28

$23-36=$ Find the critical numbers of the function.
$$g(t)=|3 t-4|$$

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Problem 29

$23-36=$ Find the critical numbers of the function.
$$g(y)=\frac{y-1}{y^{2}-y+1}$$

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Problem 30

$23-36=$ Find the critical numbers of the function.
$$h(p)=\frac{p-1}{p^{2}+4}$$

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Problem 31

$23-36=$ Find the critical numbers of the function.
$$F(x)=x^{4 / 5}(x-4)^{2}$$

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Problem 32

$23-36=$ Find the critical numbers of the function.
$$g(x)=x^{1 / 3}-x^{-2 / 3}$$

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Problem 33

$23-36=$ Find the critical numbers of the function.
$$f(\theta)=2 \cos \theta+\sin ^{2} \theta$$

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Problem 34

$23-36=$ Find the critical numbers of the function.
$$g(\theta)=4 \theta-\tan \theta$$

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Problem 35

$23-36=$ Find the critical numbers of the function.
$$f(x)=x^{2} e^{-3 x}$$

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Problem 36

$23-36=$ Find the critical numbers of the function.
$$f(x)=x^{-2} \ln x$$

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Problem 37

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=12+4 x-x^{2}, \quad[0,5]$$

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Problem 38

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=5+54 x-2 x^{3}, \quad[0,4]$$

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Problem 39

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=2 x^{3}-3 x^{2}-12 x+1, \quad[-2,3]$$

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Problem 40

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=x^{3}-6 x^{2}+5, \quad[-3,5]$$

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Problem 41

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=3 x^{4}-4 x^{3}-12 x^{2}+1, \quad[-2,3]$$

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Problem 42

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=\left(x^{2}-1\right)^{3}, \quad[-1,2]$$

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Problem 43

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(t)=t \sqrt{4-t^{2}}, \quad[-1,2]$$

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Problem 44

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=\frac{x}{x^{2}-x+1},[0,3]$$

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Problem 45

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(t)=2 \cos t+\sin 2 t, \quad[0, \pi / 2]$$

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Problem 46

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(t)=t+\cot \left(\frac{1}{2} t\right), \quad[\pi / 4,7 \pi / 4]$$

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Problem 47

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=x e^{-x^{2} / 8}, \quad[-1,4]$$

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Problem 48

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=x-\ln x, \quad\left[\frac{1}{2}, 2\right]$$

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Problem 49

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=\ln \left(x^{2}+x+1\right), \quad[-1,1]$$

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Problem 50

$37-50=$ Find the absolute maximum and absolute minimum
values of $f$ on the given interval.
$$f(x)=x-2 \tan ^{-1} x, \quad[0,4]$$

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Problem 51

If $a$ and $b$ are positive numbers, find the maximum value
of $f(x)=x^{a}(1-x)^{b}, 0 \leqslant x \leqslant 1$

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Problem 52

Use a graph to estimate the critical numbers of
$f(x)=\left|x^{3}-3 x^{2}+2\right|$ correct to one decimal place.

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Problem 53

$53-56$
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
$$f(x)=x^{5}-x^{3}+2, \quad-1 \leqslant x \leqslant 1$$

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Problem 54

$53-56$
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
$$f(x)=e^{x}+e^{-2 x}, \quad 0 \leqslant x \leqslant 1$$

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Problem 55

$53-56$
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
$$f(x)=x \sqrt{x-x^{2}}$$

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Problem 56

$53-56$
(a) Use a graph to estimate the absolute maximum and
minimum values of the function to two decimal places.
(b) Use calculus to find the exact maximum and minimum
values.
$$f(x)=x-2 \cos x, \quad-2 \leqslant x \leqslant 0$$

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Problem 57

Between $0^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C}$ , the volume $V$ (in cubic centimeters of 1 $\mathrm{kg}$ of water at a temperature $T$ is given
approximately by the formula
$$V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{3}$$
Find the temperature at which water has its maximum
density.

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Problem 58

An object with weight $W$ is dragged along a horizontal
plane by a force acting along a rope attached to the object.
If the rope makes an angle $\theta$ with the plane, then the magnitude of the force is
$$F=\frac{\mu W}{\mu \sin \theta+\cos \theta}$$
where $\mu$ is a positive constant called the coefficient of friction and where 0$\leqslant \theta \leqslant \pi / 2 .$ Show that $F$ is minimized
when $\tan \theta=\mu$

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Problem 59

A model for the US average price of a pound of white sugar
from 1993 to 2003 is given by the function
$$S(t)=-0.00003237 t^{5}+0.0009037 t^{4}-0.008956 t^{3}
+0.03629 t^{2}-0.04458 t+0.4074$$
where $t$ is measured in years since August of $1993 .$ Estimate
the times when sugar was cheapest and most expensive during the period $1993-2003 .$

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Problem 60

The Hubble Space Telescope was deployed April $24,1990,$
by the space shuttle Discovery. A model for the velocity of
the shuttle during this mission, from liftoff at $t=0$ until
the solid rocket boosters were jettisoned at $t=126 \mathrm{s},$ is
given by
$$v(t)=0.001302 t^{3}-0.09029 t^{2}+23.61 t-3.083$$
(in feet per second). Using this model, estimate the absolute
maximum and minimum values of the acceleration of the
shuttle between liftoff and the jettisoning of the boosters.

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Problem 61

When a foreign object lodged in the trachea (windpipe)
forces a person to cough, the diaphragm thrusts upward
causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower
channel for the expelled air to flow through. For a given
amount of air to escape in a fixed time, it must move faster
through the narrower channel than the wider one. The
greater the velocity of the airstream, the greater the force on
the foreign object. $X$ -rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal
radius during a cough. According to a mathematical model
of coughing, the velocity $v$ of the airstream is related to the
radius $r$ of the trachea by the equation
$$v(r)=k\left(r_{0}-r\right) r^{2} \quad \frac{1}{2} r_{0} \leqslant r \leqslant r_{0}$$
where $k$ is a constant and $r_{0}$ is the normal radius of the trachea. The restriction on $r$ is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than
$\frac{1}{2} r_{0}$ is prevented (otherwise the person would suffocate).
(a) Determine the value of $r$ in the interval $\left[\frac{1}{2} r_{0}, r_{0}\right]$ at
which $v$ has an absolute maximum. How does this
compare with experimental evidence?
(b) What is the absolute maximum value of $v$ on the interval?
(c) Sketch the graph of $v$ on the interval $\left[0, r_{0}\right] .$

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Problem 62

Show that 5 is a critical number of the function
$g(x)=2+(x-5)^{3}$ but $g$ does not have a local extreme
value at $5 .$

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Problem 63

Prove that the function $f(x)=x^{101}+x^{51}+x+1$ has
neither a local maximum nor a local minimum.

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Problem 64

If $f$ has a local minimum value at $c,$ show that the function
$g(x)=-f(x)$ has a local maximum value at $c .$

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Problem 65

Prove Fermat's Theorem for the case in which $f$ has a local
minimum at $c .$

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Problem 66

A cubic function is a polynomial of degree $3 ;$ that is, it has
the form $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a \neq 0$ .
(a) Show that a cubic function can have two, one, or no
critical number(s). Give examples and sketches to illus-
trate the three possibilities.
(b) How many local extreme values can a cubic function
have?

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