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Chapter 4

Applications of Differentiation

Educators


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 $[a, b] .$
$$
\begin{array}{l}{\text { (a) What theorem guarantees the existence of an absolute max- }} \\ {\text { imum value and an absolute minimum value for } f ?} \\ {\text { (b) What steps would you take to find those maximum and }} \\ {\text { minimum values? }}\end{array}
$$

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Problem 3

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 min-
imum, a local maximum or minimum, or neither a maximum
nor a minimum.

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Problem 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 min-
imum, a local maximum or minimum, or neither a maximum
nor a minimum.

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Problem 5

Use the graph to state the absolute and local maximum and
minimum values of the function.

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Problem 6

Use the graph to state the absolute and local maximum and
minimum values of the function.

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Problem 7

Sketch the graph of a function $f$ that is continuous on $[1,5]$
and has the given properties.
$$
\begin{array}{l}{\text { Absolute minimum at } 2, \text { absolute maximum at } 3,} \\ {\text { local minimum at } 4}\end{array}
$$

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Problem 8

$$
\begin{array}{l}{\text { Absolute minimum at } 1, \text { absolute maximum at } 5 \text { , }} \\ {\text { local maximum at } 2, \text { local minimum at } 4}\end{array}
$$

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Problem 9

Absolute maximum at $5,$ absolute minimum at 2 ,
local maximum at $3,$ local minima at 2 and 4

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Problem 10

f has no local maximum or minimum, but 2 and 4 are critical
numbers

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Problem 11

$$
\begin{array}{l}{\text { (a) Sketch the graph of a function that has a local maximum }} \\ {\text { at } 2 \text { and is differentiable at } 2 \text { . }} \\ {\text { (b) Sketch the graph of a function that has a local maximum }} \\ {\text { at } 2 \text { and is continuous but not differentiable at } 2 \text { . }}\end{array}
$$
$$
\begin{array}{l}{\text { (c) Sketch the graph of a function that has a local maximum }} \\ {\text { at } 2 \text { and is not continuous at } 2 .}\end{array}
$$

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Problem 12

$$
\begin{array}{l}{\text { (a) Sketch the graph of a function on }[-1,2] \text { that has an }} \\ {\text { absolute maximum but no local maximum. }} \\ {\text { (b) Sketch the graph of a function on }[-1,2] \text { that has a local }} \\ {\text { maximum but no absolute maximum. }}\end{array}
$$

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Problem 13

$$
\begin{array}{l}{\text { (a) Sketch the graph of a function on }[-1,2] \text { that has an }} \\ {\text { absolute maximum but no absolute minimum. }} \\ {\text { (b) Sketch the graph of a function on }[-1,2] \text { that is discontin- }} \\ {\text { uous but has both an absolute maximum and an absolute }} \\ {\text { minimum. }}\end{array}
$$

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Problem 14

$$
\begin{array}{l}{\text { (a) Sketch the graph of a function that has two local maxima, }} \\ {\text { one local minimum, and no absolute minimum. }} \\ {\text { (b) Sketch the graph of a function that has three local minima, }} \\ {\text { two local maxima, and seven critical numbers. }}\end{array}
$$

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Problem 15

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 and $1.3 . )$
$$f(x)=\frac{1}{2}(3 x-1), \quad x \leqslant 3$$

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Problem 16

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 and $1.3 . )$
$$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$$

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Problem 17

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 and $1.3 . )$
$$f(x)=1 / x, \quad x \geqslant 1$$

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Problem 18

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 and $1.3 . )$
$$f(x)=1 / x, \quad 1<x<3$$

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Problem 19

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 and $1.3 . )$
$$f(x)=\sin x, \quad 0 \leqslant x<\pi / 2$$

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Problem 20

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 and $1.3 . )$
$$f(x)=\sin x, \quad 0<x \leqslant \pi / 2$$

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Problem 21

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 and $1.3 . )$
$$f(x)=\sin x, \quad-\pi / 2 \leqslant x \leqslant \pi / 2$$

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Problem 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 and $1.3 . )$
$$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

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Problem 23

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 and $1.3 . )$
$$f(x)=\ln x$ $0<x \leqslant 2$$

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Problem 24

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 and $1.3 . )$
$$f(x)=|x|$$

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Problem 25

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 and $1.3 . )$
$$f(x)=1-\sqrt{x}$$

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Problem 26

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 and $1.3 . )$
$$f(x)=e^{x}$$

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Problem 27

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 and $1.3 . )$
$f(x)=\left\{\begin{array}{ll}{1-x} & {\text { if } 0 \leqslant x<2} \\ {2 x-4} & {\text { if } 2 \leq x \leq 3}\end{array}\right.$

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Problem 28

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 and $1.3 . )$
$f(x)=\left\{\begin{array}{ll}{4-x^{2}} & {\text { if }-2 \leq x<0} \\ {2 x-1} & {\text { if } 0 \leqslant x \leqslant 2}\end{array}\right.$

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Problem 29

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

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Problem 30

$29-44$ Find the critical numbers of the function.
$$f(x)=x^{3}+6 x^{2}-15 x$$

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Problem 31

$29-44$ Find the critical numbers of the function.
$$f(x)=2 x^{3}-3 x^{2}-36 x$$

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Problem 32

$29-44$ Find the critical numbers of the function.
$$f(x)=2 x^{3}+x^{2}+2 x$$

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Problem 33

$29-44$ Find the critical numbers of the function.
$$g(t)=t^{4}+t^{3}+t^{2}+1$$

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Problem 34

$29-44$ Find the critical numbers of the function.
$$g(t)=|3 t-4|$$

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Problem 35

$29-44$ Find the critical numbers of the function.
$$g(y)=\frac{y-1}{y^{2}-y+1}$$

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Problem 36

$29-44$ Find the critical numbers of the function.
$$h(p)=\frac{p-1}{p^{2}+4}$$

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Problem 37

$29-44$ Find the critical numbers of the function.
$$h(t)=t^{3 / 4}-2 t^{1 / 4}$$

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Problem 38

$29-44$ Find the critical numbers of the function.
$$g(x)=x^{1 / 3}-x^{-2 / 3}$$

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Problem 39

$29-44$ Find the critical numbers of the function.
$$F(x)=x^{4 / 5}(x-4)^{2}$$

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Problem 40

$29-44$ Find the critical numbers of the function.
$$g(\theta)=4 \theta-\tan \theta$$

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Problem 41

$29-44$ Find the critical numbers of the function.
$$f(\theta)=2 \cos \theta+\sin ^{2} \theta$$

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Problem 42

$29-44$ Find the critical numbers of the function.
$$h(t)=3 t-\arcsin t$$

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Problem 43

$29-44$ Find the critical numbers of the function.
$$f(x)=x^{2} e^{-3 x}$$

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Problem 44

$29-44$ Find the critical numbers of the function.
$$f(x)=x^{-2} \ln x$$

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Problem 45

A formula for the derivative of a function $f$ is given. How many critical numbers does f have?
$$f^{\prime}(x)=5 e^{-0.1|x|} \sin x-1$$

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Problem 46

A formula for the derivative of a function $f$ is given. How many critical numbers does f have?
$$f^{\prime}(x)=\frac{100 \cos ^{2} x}{10+x^{2}}-1$$

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Problem 47

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

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Problem 48

$47-62$ 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 49

$47-62$ Find the absolute maximum and absolute minimum values
of $f$ on the given interval.
$$f(x)=2 x^{3}-3 x^{2}-12 x+1,[-2,3]$$

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Problem 50

$47-62$ 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 51

$47-62$ Find the absolute maximum and absolute minimum values
of $f$ on the given interval.
$$f(x)=3 x^{4}-4 x^{3}-12 x^{2}+1,[-2,3]$$

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Problem 52

$47-62$ Find the absolute maximum and absolute minimum values
of $f$ on the given interval.
$$f(x)=x+\frac{1}{x}, \quad[0.2,4]$$

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Problem 53

$47-62$ Find the absolute maximum and absolute minimum values of $f$ on the given interval.
$$f(x)=x+\frac{1}{x}, \quad[0.2,4]$$

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Problem 54

$47-62$ 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 55

$47-62$ 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 56

$47-62$ Find the absolute maximum and absolute minimum values of $f$ on the given interval.
$$f(t)=\sqrt[3]{t}(8-t), \quad[0,8]$$

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Problem 57

$47-62$ 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 58

$47-62$ Find the absolute maximum and absolute minimum values of $f$ on the given interval.
$$f(t)=t+\cot (t / 2), \quad[\pi / 4,7 \pi / 4]$$

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Problem 59

$47-62$ 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 60

$47-62$ 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 61

$47-62$ 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 62

$47-62$ 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 63

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

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Problem 64

64. 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 65

(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.
$$f(x)=x^{5}-x^{3}+2, \quad-1 \leq x \leq 1$$

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Problem 66

(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.
$$f(x)=e^{x}+e^{-2 x}, 0 \leqslant x \leqslant 1$$

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Problem 67

(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 68

(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.
$$f(x)=x-2 \cos x, \quad-2 \leqq x \leqslant 0$$

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Problem 69

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
$$
V=999.87-0.06426 T+0.0085043 T^{2}-0.0006679 T^{3}
$$
Find the temperature at which water has its maximum
density.

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Problem 70

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 71

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

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Problem 72

On May $7,1992,$ the space shuttle Endeavour was launched
on mission STS-49, the purpose of which was to install a
new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between
liftoff and the jettisoning of the solid rocket boosters.
$$
\begin{array}{l}{\text { (a) Use a graphing calculator or computer to find the cubic }} \\ {\text { polynomial that best models the velocity of the shuttle for }} \\ {\text { the time interval } t \in[0,125] . \text { Then graph this polynomial. }}\end{array}
$$
$$
\begin{array}{l}{\text { (b) Find a model for the acceleration of the shuttle and use it }} \\ {\text { to estimate the maximum and minimum values of the }} \\ {\text { acceleration during the first } 125 \text { seconds. }}\end{array}
$$

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Problem 73

When a foreign object lodged in the trachea (windpipe)
forces a person to cough, the diaphragm thrusts upward caus-
ing 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 $r$ 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).
$$
\begin{array}{l}{\text { (a) Determine the value of } r \text { in the interval }\left[\frac{1}{2} r_{0}, r_{0}\right] \text { at which }} \\ {v \text { has an absolute maximum. How does this compare with }} \\ {\text { experimental evidence? }}\end{array}
$$
$$
\begin{array}{l}{\text { (b) What is the absolute maximum value of } v \text { on the interval? }} \\ {\text { (c) Sketch the graph of } v \text { on the interval }\left[0, r_{0}\right] .}\end{array}
$$

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Problem 74

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

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Problem 75

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 76

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 77

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

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Problem 78

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 $a \neq 0$
$$
\begin{array}{l}{\text { (a) Show that a cubic function can have two, one, or no criti- }} \\ {\text { cal number(s). Give examples and sketches to illustrate }} \\ {\text { the three possibilities. }} \\ {\text { (b) How many local extreme values can a cubic function }} \\ {\text { have? }}\end{array}
$$

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