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 .

Doruk I.

Numerade Educator

any absolute extreme values on $[a, b]$ . Then explain how your

answer is consistent with Theorem 1 .

Matt J.

Numerade Educator

any absolute extreme values on $[a, b]$ . Then explain how your

answer is consistent with Theorem 1 .

Doruk I.

Numerade Educator

any absolute extreme values on $[a, b]$ . Then explain how your

answer is consistent with Theorem 1 .

Matt J.

Numerade Educator

any absolute extreme values on $[a, b]$ . Then explain how your

answer is consistent with Theorem 1 .

Doruk I.

Numerade Educator

any absolute extreme values on $[a, b]$ . Then explain how your

answer is consistent with Theorem 1 .

Matt J.

Numerade Educator

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

occur.

Doruk I.

Numerade Educator

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

occur.

Matt J.

Numerade Educator

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

occur.

Doruk I.

Numerade Educator

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

occur.

Matt J.

Numerade Educator

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

$$\begin{array}{l}{x} && {f^{\prime}(x)} \\ {a} && {0} \\ {b} && {0} \\ {c} && {5}\end{array}$$

Doruk I.

Numerade Educator

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

$$

\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & {0} \\ {b} & {0} \\ {c} & {-5}\end{array}

$$

Matt J.

Numerade Educator

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

$$

\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & \text{does not exist}\\ {b} & {0} \\ {c} & {-2}\end{array}

$$

Doruk I.

Numerade Educator

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

$$

\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & \text{does not exist}\\ {b} & \text{does not exist} \\ {c} & {-1.7}\end{array}

$$

Matt J.

Numerade Educator

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

$$

Doruk I.

Numerade Educator

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

$$

Matt J.

Numerade Educator

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 I.

Numerade Educator

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 J.

Numerade Educator

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

$$

Doruk I.

Numerade Educator

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}{ll}{x+1,} & {-1 \leq x<0} \\ {\cos x,} & {0 < x \leq \frac{\pi}{2}}\end{array}\right.

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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}, \quad-1 \leq x \leq 8

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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}}, \quad-2 \leq x \leq 1

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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}

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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}

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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}

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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}

$$

Matt J.

Numerade Educator

In Exercises $21-36,$ 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

$$

Doruk I.

Numerade Educator

In Exercises $21-36,$ 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

$$

Matt J.

Numerade Educator

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.

$$

f(x)=x^{4 / 3},-1 \leq x \leq 8

$$

Doruk I.

Numerade Educator

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.

$$

f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8

$$

Matt J.

Numerade Educator

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.

$$

g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1

$$

Doruk I.

Numerade Educator

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.

$$

h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8

$$

Matt J.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

y=x^{2}-6 x+7

$$

Doruk I.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

f(x)=6 x^{2}-x^{3}

$$

Matt J.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

f(x)=x(4-x)^{3}

$$

Doruk I.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

g(x)=(x-1)^{2}(x-3)^{2}

$$

Matt J.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

y=x^{2}+\frac{2}{x}

$$

Doruk I.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

f(x)=\frac{x^{2}}{x-2}

$$

Matt J.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

y=x^{2}-32 \sqrt{x}

$$

Doruk I.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

g(x)=\sqrt{2 x-x^{2}}

$$

Matt J.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

y=x^{3}+3 x^{2}-24 x+7

$$

Doruk I.

Numerade Educator

In Exercises $41-50,$ determine all critical points for each function.

$$

y=x-3 x^{2 / 3}

$$

Matt J.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=x^{2 / 3}(x+2)

$$

Doruk I.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=x^{2 / 3}\left(x^{2}-4\right)

$$

Matt J.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=x \sqrt{4-x^{2}}

$$

Doruk I.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=x^{2} \sqrt{3-x}

$$

Matt J.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=\left\{\begin{array}{ll}{4-2 x,} & {x \leq 1} \\ {x+1,} & {x>1}\end{array}\right.

$$

Doruk I.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=\left\{\begin{array}{ll}{3-x,} & {x<0} \\ {3+2 x-x^{2},} & {x \geq 0}\end{array}\right.

$$

Matt J.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

y=\left\{\begin{array}{ll}{-x^{2}-2 x+4,} & {x \leq 1} \\ {-x^{2}+6 x-4,} & {x>1}\end{array}\right.

$$

Doruk I.

Numerade Educator

In Exercises $51-58$ , find the critical points and domain endpoints for

each function. Then find the value of the function at each of these

points and identify extreme values (absolute and local).

$$

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.

$$

Matt J.

Numerade Educator

In Exercises 59 and $60,$ give reasons for your answers.

Let

$$

f(x)=(x-2)^{2 / 3}

$$

$$

\begin{array}{l}{\text { a. Does } f^{\prime}(2) \text { exist? }} \\ {\text { b. Show that the only local extreme value of } f \text { occurs at } x=2 \text { . }} \\ {\text { c. Does the result in part (b) contradict the Extreme Value }} \\ {\text { Theorem? }} \\ {\text { d. Repeat parts (a) and (b) for } f(x)=(x-a)^{2 / 3}, \text { replacing } 2 \text { by } a \text { . }}\end{array}

$$

Doruk I.

Numerade Educator

In Exercises 59 and $60,$ give reasons for your answers.

Let

$$f(x)=\left|x^{3}-9 x\right|$$

$$\begin{array}{ll}{\text { a. Does } f^{\prime}(0) \text { exist? }} & {\text { b. Does } f^{\prime}(3) \text { exist? }} \\ {\text { c. Does } f^{\prime}(-3) \text { exist? }} & {\text { d. Determine all extrema of } f}\end{array}$$

Matt J.

Numerade Educator

In Exercises $61-62,$ show that the function has neither an absolute

minimum nor an absolute maximum on its natural domain.

$$

y=x^{11}+x^{3}+x-5

$$

Doruk I.

Numerade Educator

In Exercises $61-62,$ show that the function has neither an absolute

minimum nor an absolute maximum on its natural domain.

$$

y=3 x+\tan x

$$

Matt J.

Numerade Educator

A minimum with no derivative The function $f(x)=|x|$ has an

absolute minimum value at $x=0$ even though $f$ is not differen-

tiable at $x=0 .$ Is this consistent with Theorem 2$?$ Give reasons

for your answer.

Doruk I.

Numerade Educator

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.

Matt J.

Numerade Educator

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.

Doruk I.

Numerade Educator

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.

Matt J.

Numerade Educator

The function

$$V(x)=x(10-2 x)(16-2 x), \quad 0 < x < 5$$

$$\begin{array}{l}{\text { models the volume of a box. }} \\ {\text { a. Find the extreme values of } V \text { . }} \\ {\text { b. Interpret any values found in part (a) in terms of the volume }} \\ {\text { of the box. }}\end{array}$$

Doruk I.

Numerade Educator

Cubic functions Consider the cubic function

$$f(x)=a x^{3}+b x^{2}+c x+d$$

$$\begin{array}{l}{\text { a. Show that } f \text { can have } 0,1, \text { or } 2 \text { critical points. Give examples }} \\ {\text { and graphs to support your argument. }} \\ {\text { b. How many local extreme values can } f \text { have? }}\end{array}$$

Matt J.

Numerade Educator

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.

Doruk I.

Numerade Educator

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)?

Matt J.

Numerade Educator

Graph the functions in Exercises $71-74 .$ Then find the extreme values

of the function on the interval and say where they occur.

$$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

Doruk I.

Numerade Educator

Graph the functions in Exercises $71-74 .$ 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

$$

Matt J.

Numerade Educator

Graph the functions in Exercises $71-74 .$ 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

$$

Doruk I.

Numerade Educator

Graph the functions in Exercises $71-74 .$ 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

$$

Matt J.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]

$$

Doruk I.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]

$$

Matt J.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

f(x)=x^{2 / 3}(3-x), \quad[-2,2]

$$

Doruk I.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

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

$$

Matt J.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]

$$

Doruk I.

Numerade Educator

In Exercises $75-80,$ you will use a CAS to help find the absolute

extrema of the given function over the specified closed interval. Perform

the following steps.

$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$

$$

f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]

$$

Matt J.

Numerade Educator