Thomas Calculus

George B. Thomas, Jr.

Chapter 4

Applications of Derivatives - all with Video Answers

Educators

+ 6 more educators

Section 1

Extreme Values of Functions

Problem 1

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Numerade Educator

Problem 3

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Numerade Educator

Problem 4

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Numerade Educator

Problem 5

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Numerade Educator

Problem 6

In Exercises $1-6,$ determine from the graph whether the function has
any absolute extreme values on $[a, b] .$ Then explain how your
answer is consistent with. Theorem 1

Carson Merrill

Numerade Educator

Problem 7

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

Stephen Hobbs

Numerade Educator

Problem 8

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

Stephen Hobbs

Numerade Educator

Problem 9

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

Stephen Hobbs

Numerade Educator

Problem 10

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

Stephen Hobbs

Numerade Educator

Problem 11

In Exercises $11-14,$ match the table with a graph.

Doruk Isik

Numerade Educator

Problem 12

In Exercises $11-14,$ match the table with a graph.

Doruk Isik

Numerade Educator

Problem 13

In Exercises $11-14,$ match the table with a graph.

Doruk Isik

Numerade Educator

Problem 14

In Exercises $11-14,$ match the table with a graph.

Doruk Isik

Numerade Educator

Problem 15

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
f(x)=|x|, \quad-1<x<2
$$

Matt Just

Numerade Educator

Problem 16

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
y=\frac{6}{x^{2}+2}, \quad-1<x<1
$$

Doruk Isik

Numerade Educator

Problem 17

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
g(x)=\left\{\begin{array}{ll}{-x,} & {0 \leq x<1} \\ {x-1,} & {1 \leq x \leq 2}\end{array}\right.
$$

Doruk Isik

Numerade Educator

Problem 18

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
h(x)=\left\{\begin{array}{ll}{\frac{1}{x},} & {-1 \leq x<0} \\ {\sqrt{x},} & {0 \leq x \leq 4}\end{array}\right.
$$

Matt Just

Numerade Educator

Problem 19

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
y=3 \sin x, \quad 0<x<2 \pi
$$

Matt Just

Numerade Educator

Problem 20

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem 1 .
$$
f(x)=\left\{\begin{array}{lr}{x+1,} & {-1 \leq x<0} \\ {\cos x,} & {0<x \leq \frac{\pi}{2}}\end{array}\right.
$$

Matt Just

Numerade Educator

Problem 21

In Exercises $21-40,$ find the absolute maximum and minimum values
of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3
$$

Natalie Belcher

Natalie Belcher

Numerade Educator

Problem 22

In Exercises $21-40,$ find the absolute maximum and minimum values
of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(x)=-x-4, \quad-4 \leq x \leq 1
$$

Jacob Fry

Numerade Educator

Problem 23

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(x)=x^{2}-1, \quad-1 \leq x \leq 2
$$

Jacob Fry

Numerade Educator

Problem 24

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(x)=4-x^{3}, \quad-2 \leq x \leq 1
$$

Jacob Fry

Numerade Educator

Problem 25

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2
$$

Jacob Fry

Numerade Educator

Problem 26

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
F(x)=-\frac{1}{x}, \quad-2 \leq x \leq-1
$$

Jacob Fry

Numerade Educator

Problem 27

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
h(x)=\sqrt[3]{x},-1 \leq x \leq 8
$$

Nicole Hoffman

Numerade Educator

Problem 28

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
h(x)=-3 x^{2 / 3}, \quad-1 \leq x \leq 1
$$

Nicole Hoffman

Numerade Educator

Problem 29

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
g(x)=\sqrt{4-x^{2}},-2 \leq x \leq 1
$$

Nicole Hoffman

Numerade Educator

Problem 30

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
g(x)=-\sqrt{5-x^{2}}, \quad-\sqrt{5} \leq x \leq 0
$$

Nicole Hoffman

Numerade Educator

Problem 31

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}
$$

Kim Matthews

Numerade Educator

Problem 32

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}
$$

Kim Matthews

Numerade Educator

Problem 32

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}
$$

Kim Matthews

Numerade Educator

Problem 33

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}
$$

Kim Matthews

Numerade Educator

Problem 34

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
g(x)=\sec x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{6}
$$

Kim Matthews

Numerade Educator

Problem 35

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(t)=2-|t|, \quad-1 \leq t \leq 3
$$

Jackie Stone

Numerade Educator

Problem 36

In Exercises $21-40,$ find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$
f(t)=|t-5|, \quad 4 \leq t \leq 7
$$

Jackie Stone

Numerade Educator

Problem 37

In Exercises $37-40,$ find the function's absolute maximum and minimum values and say where they are assumed.
$$
f(x)=x^{4 / 3}, \quad-1 \leq x \leq 8
$$

Doruk Isik

Numerade Educator

Problem 38

In Exercises $37-40,$ find the function's absolute maximum and minimum values and say where they are assumed.
$$
f(x)=x^{5 / 3},-1 \leq x \leq 8
$$

Matt Just

Numerade Educator

Problem 39

In Exercises $37-40,$ find the function's absolute maximum and minimum values and say where they are assumed.
$$
g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1
$$

Doruk Isik

Numerade Educator

Problem 40

In Exercises $37-40,$ find the function's absolute maximum and mini-
mum values and say where they are assumed.
$$
h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 41

In Exercises $41-48,$ determine all critical points for each function.
$$
y=x^{2}-6 x+7
$$

Doruk Isik

Numerade Educator

Problem 42

In Exercises $41-48,$ determine all critical points for each function.
$$
f(x)=6 x^{2}-x^{3}
$$

Anh Hoang

Numerade Educator

Problem 43

In Exercises $41-48,$ determine all critical points for each function.
$$
f(x)=x(4-x)^{3}
$$

Anh Hoang

Numerade Educator

Problem 44

In Exercises $41-48,$ determine all critical points for each function.
$$
g(x)=(x-1)^{2}(x-3)^{2}
$$

Matt Just

Numerade Educator

Problem 44

In Exercises $41-48,$ determine all critical points for each function.
$$
g(x)=(x-1)^{2}(x-3)^{2}
$$

Matt Just

Numerade Educator

Problem 45

In Exercises $41-48,$ determine all critical points for each function.
$$
y=x^{2}+\frac{2}{x}
$$

Doruk Isik

Numerade Educator

Problem 46

In Exercises $41-48,$ determine all critical points for each function.
$$
f(x)=\frac{x^{2}}{x-2}
$$

Anh Hoang

Numerade Educator

Problem 47

In Exercises $41-48,$ determine all critical points for each function.
$$
y=x^{2}-32 \sqrt{x}
$$

Doruk Isik

Numerade Educator

Problem 48

In Exercises $41-48,$ determine all critical points for each function.
$$
g(x)=\sqrt{2 x-x^{2}}
$$

Matt Just

Numerade Educator

Problem 49

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=2 x^{2}-8 x+9
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 50

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=x^{3}-2 x+4
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 51

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=x^{3}+x^{2}-8 x+5
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 52

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=x^{3}(x-5)^{2}
$$

Melissa Munoz

Numerade Educator

Problem 53

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=\sqrt{x^{2}-1}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 54

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=x-4 \sqrt{x}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 55

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=\frac{1}{\sqrt[3]{1-x^{2}}}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 56

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=\sqrt{3+2 x-x^{2}}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 57

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=\frac{x}{x^{2}+1}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 58

In Exercises $49-58$ , find the extreme values (absolute and local) of the
function over its natural domain, and where they occur.
$$
y=\frac{x+1}{x^{2}+2 x+2}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 59

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=x^{2 / 3}(x+2)
$$

Anh Hoang

Numerade Educator

Problem 60

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=x^{2 / 3}\left(x^{2}-4\right)
$$

Anh Hoang

Numerade Educator

Problem 61

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=x \sqrt{4-x^{2}}
$$

Anh Hoang

Numerade Educator

Problem 62

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=x^{2} \sqrt{3-x}
$$

Anh Hoang

Numerade Educator

Problem 63

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=\left\{\begin{array}{ll}{4-2 x,} & {x \leq 1} \\ {x+1,} & {x>1}\end{array}\right.
$$

Anh Hoang

Numerade Educator

Problem 64

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=\left\{\begin{array}{ll}{3-x,} & {x<0} \\ {3+2 x-x^{2},} & {x \geq 0}\end{array}\right.
$$

Anh Hoang

Numerade Educator

Problem 65

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=\left\{\begin{array}{ll}{-x^{2}-2 x+4,} & {x \leq 1} \\ {-x^{2}+6 x-4,} & {x>1}\end{array}\right.
$$

Carson Merrill

Numerade Educator

Problem 66

In Exercises $59-66,$ find the critical points, domain endpoints, and
extreme values (absolute and local) for each function.
$$
y=\left\{\begin{array}{ll}{-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4},} & {x \leq 1} \\ {x^{3}-6 x^{2}+8 x,} & {x>1}\end{array}\right.
$$

Carson Merrill

Numerade Educator

Problem 67

In Exercises 67 and $68,$ give reasons for your answers.
Let $f(x)=(x-2)^{2 / 3}$
a. Does $f^{\prime}(2)$ exist?
b. Show that the only local extreme value of $f$ occurs at $x=2$ .
c. Does the result in part (b) contradict the Extreme Value Theorem?
d. Repeat parts (a) and (b) for $f(x)=(x-a)^{2 / 3},$ replacing 2 by $a .$

Carson Merrill

Numerade Educator

Problem 68

In Exercises 67 and $68,$ give reasons for your answers.
Let $f(x)=\left|x^{3}-9 x\right|$
$\begin{array}{ll}{\text { a. Does } f^{\prime}(0) \text { exist? }} & {\text { b. Does } f^{\prime}(3) \text { exist? }} \\ {\text { c. Does } f^{\prime}(-3) \text { exist? }} & {\text { d. Determine all extrema of } f}\end{array}$

Carson Merrill

Numerade Educator

Problem 69

A minimum with no derivative The function $f(x)=|x|$ has
an absolute minimum value at $x=0$ even though $f$ is not differentiable at $x=0 .$ Is this consistent with Theorem 2$?$ Give reasons for your answer.

Carson Merrill

Numerade Educator

Problem 70

Even functions If an even function $f(x)$ has a local maximum
value at $x=c,$ can anything be said about the value of $f$ at
$x=-c ?$ Give reasons for your answer.

Mj Santos

Numerade Educator

Problem 71

Odd functions If an odd function $g(x)$ has a local minimum
value at $x=c,$ can anything be said about the value of $g$ at
$x=-c ?$ Give reasons for your answer.

Nick Johnson

Numerade Educator

Problem 72

No critical points or endpoints exist We know how to find the
extreme values of a continuous function $f(x)$ by investigating its
values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions
really exist? Give reasons for your answers.

Seth Gerberding

Seth Gerberding

Numerade Educator

Problem 73

The function
$$V(x)=x(10-2 x)(16-2 x), \quad 0< x <5$$
models the volume of a box.
a. Find the extreme values of $V$ .
b. Interpret any values found in part (a) in terms of the volume
of the box.

Doruk Isik

Numerade Educator

Problem 74

Cubic functions Consider the cubic function
$$
s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, \quad g>0
$$
a. Show that $f$ can have $0,1,$ or 2 critical points. Give examples
and graphs to support your argument.
b. How many local extreme values can $f$ have?

Carson Merrill

Numerade Educator

Problem 75

Maximum height of a vertically moving body The height of a
body moving vertically is given by
$$
s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}, \quad g>0
$$
with $s$ in meters and $t$ in seconds. Find the body's maximum height.

Mutahar Mehkri

Numerade Educator

Problem 76

Peak alternating current Suppose that at any given time $t$ (in
seconds) the current $i$ (in amperes) in an alternating current circuit is $i=2 \cos t+2 \sin t .$ What is the peak current for this circuit (largest magnitude)?

Khushbu Rani

Numerade Educator

Problem 77

Graph the functions in Exercises $77-80 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$
f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5
$$

WM

William Mead

Numerade Educator

Problem 78

Graph the functions in Exercises $77-80 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$
g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7
$$

Carson Merrill

Numerade Educator

Problem 79

Graph the functions in Exercises $77-80 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$
h(x)=|x+2|-|x-3|, \quad-\infty<x<\infty
$$

Gaurav Kalra

Numerade Educator

Problem 80

Graph the functions in Exercises $77-80 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$
k(x)=|x+1|+|x-3|, \quad-\infty<x<\infty
$$

Gaurav Kalra

Numerade Educator

Problem 81

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]
$$

Carson Merrill

Numerade Educator

Problem 82

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]
$$

Carson Merrill

Numerade Educator

Problem 83

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=x^{2 / 3}(3-x), \quad[-2,2]
$$

Carson Merrill

Numerade Educator

Problem 84

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]
$$

Carson Merrill

Numerade Educator

Problem 85

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]
$$

Carson Merrill

Numerade Educator

Problem 86

In Exercises $81-86,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0 .$ (In some exercises, you
may have to use the numerical equation solver to approximate a
solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at
the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and
identify where they occur.
$$
f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]
$$

Carson Merrill

Numerade Educator