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Calculus: Graphical, Numerical, Algebraic

Ross L. Finney, Franklin D. Demana, Bet K. Waits, Daniel Kennedy

Chapter 4

Applications of Derivatives - all with Video Answers

Educators

+ 1 more educators

Section 1

Extreme Values of Functions

01:37

Problem 1

In Exercises $1-4,$ find the extreme values and where they occur.

Carson Merrill
Carson Merrill
Numerade Educator
00:40

Problem 2

In Exercises $1-4,$ find the extreme values and where they occur.

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 3

In Exercises $1-4,$ find the extreme values and where they occur.

Linh Vu
Linh Vu
Numerade Educator
01:43

Problem 4

In Exercises $1-4,$ find the extreme values and where they occur.

Linda Hand
Linda Hand
Numerade Educator
01:07

Problem 5

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=h(x)$$

Linda Hand
Linda Hand
Numerade Educator
00:42

Problem 6

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=f(x)$$

Linh Vu
Linh Vu
Numerade Educator
00:34

Problem 7

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=f(x)$$

Linh Vu
Linh Vu
Numerade Educator
00:35

Problem 8

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=h(x)$$

Linh Vu
Linh Vu
Numerade Educator
00:24

Problem 9

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=g(x)$$

Linh Vu
Linh Vu
Numerade Educator
00:26

Problem 10

In Exercises $5-10$ , identify each $x$ -value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.
$$y=g(x)$$

Linh Vu
Linh Vu
Numerade Educator
03:49

Problem 11

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.

$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$

Linda Hand
Linda Hand
Numerade Educator
01:41

Problem 12

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$g(x)=e^{-x}, \quad-1 \leq x \leq 1$$

Carson Merrill
Carson Merrill
Numerade Educator
01:28

Problem 13

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$h(x)=\ln (x+1), \quad 0 \leq x \leq 3$$

Carson Merrill
Carson Merrill
Numerade Educator
01:17

Problem 14

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$k(x)=e^{-x^{2}}, \quad-\infty< x <\infty$$

Linh Vu
Linh Vu
Numerade Educator
02:54

Problem 15

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$f(x)=\sin \left(x+\frac{\pi}{4}\right), \quad 0 \leq x \leq \frac{7 \pi}{4}$$

Linh Vu
Linh Vu
Numerade Educator
03:28

Problem 16

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$g(x)=\sec x, \quad-\frac{\pi}{2} < x <\frac{3 \pi}{2}$$

Linh Vu
Linh Vu
Numerade Educator
01:12

Problem 17

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$f(x)=x^{2 / 5}, \quad-3 \leq x<1$$

Linh Vu
Linh Vu
Numerade Educator
01:56

Problem 18

In Exercises $11-18,$ use analytic methods to find the extreme values of the function on the interval and where they occur.
$$f(x)=x^{3 / 5}, \quad-2< x \leq 3$$

Linh Vu
Linh Vu
Numerade Educator
01:58

Problem 19

In Exercises $19-30$ , find the extreme values of the function and where they occur.

$y=2 x^{2}-8 x+9$

Madi Sousa
Madi Sousa
Numerade Educator
01:34

Problem 20

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=x^{3}-2 x+4$$

Linh Vu
Linh Vu
Numerade Educator
02:07

Problem 21

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=x^{3}+x^{2}-8 x+5$$

Linh Vu
Linh Vu
Numerade Educator
01:54

Problem 22

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=x^{3}-3 x^{2}+3 x-2$$

Linh Vu
Linh Vu
Numerade Educator
03:14

Problem 23

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\sqrt{x^{2}-1}$$

Linh Vu
Linh Vu
Numerade Educator
02:27

Problem 24

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{1}{x^{2}-1}$$

Linh Vu
Linh Vu
Numerade Educator
02:44

Problem 25

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{1}{\sqrt{1-x^{2}}}$$

Linh Vu
Linh Vu
Numerade Educator
02:00

Problem 26

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{1}{\sqrt[3]{1-x^{2}}}$$

Linh Vu
Linh Vu
Numerade Educator
03:35

Problem 27

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\sqrt{3+2 x-x^{2}}$$

Linh Vu
Linh Vu
Numerade Educator
02:41

Problem 28

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{3}{2} x^{4}+4 x^{3}-9 x^{2}+10$$

Linh Vu
Linh Vu
Numerade Educator
01:43

Problem 29

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{x}{x^{2}+1}$$

Linh Vu
Linh Vu
Numerade Educator
04:32

Problem 30

In Exercises $19-30$ , find the extreme values of the function and where they occur.
$$y=\frac{x+1}{x^{2}+2 x+2}$$

Linh Vu
Linh Vu
Numerade Educator
03:10

Problem 31

In Exercises $31-34,$ find the extreme values of the function on the interval and where they occur.
$$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

Linh Vu
Linh Vu
Numerade Educator
02:45

Problem 32

In Exercises $31-34,$ find the extreme values of the function on the interval and where they occur.
$$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$

Linh Vu
Linh Vu
Numerade Educator
01:48

Problem 33

In Exercises $31-34,$ find the extreme values of the function on the interval and where they occur.
$$$h(x)=|x+2|-|x-3|, \quad-\infty< x <\infty$$

Linh Vu
Linh Vu
Numerade Educator
01:35

Problem 34

In Exercises $31-34,$ find the extreme values of the function on the interval and where they occur.
$$k(x)=|x+1|+|x-3|, \quad-\infty< x <\infty$$

Linh Vu
Linh Vu
Numerade Educator
02:55

Problem 35

In Exercises $35-42,$ identify the critical point and determine the local extreme values.
$$y=x^{2 / 3}(x+2)$$

Carson Merrill
Carson Merrill
Numerade Educator
03:16

Problem 36

In Exercises $35-42,$ identify the critical point and determine the local extreme values.
$$y=x^{2 / 3}\left(x^{2}-4\right)$$

Linh Vu
Linh Vu
Numerade Educator
04:19

Problem 37

In Exercises $35-42,$ identify the critical point and determine the local extreme values.
$$y=x \sqrt{4-x^{2}}$$

Linh Vu
Linh Vu
Numerade Educator
03:52

Problem 38

In Exercises $35-42,$ identify the critical point and determine the local extreme values.
$$y=x^{2} \sqrt{3-x}$$

Linh Vu
Linh Vu
Numerade Educator
01:14

Problem 39

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

Linh Vu
Linh Vu
Numerade Educator
02:24

Problem 40

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

Linh Vu
Linh Vu
Numerade Educator
02:03

Problem 41

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

Linh Vu
Linh Vu
Numerade Educator
04:06

Problem 42

$y=\left\{\begin{array}{ll}{-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4},} & {x \leq 1} \\ {x^{3}-6 x^{2}+8 x,} & {x>1}\end{array}\right.$

Linh Vu
Linh Vu
Numerade Educator
04:01

Problem 43

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 (a) in terms of volume of the box.

Linh Vu
Linh Vu
Numerade Educator
02:10

Problem 44

The function
$P(x)=2 x+\frac{200}{x}, \quad 0< x <\infty$
models the perimeter of a rectangle of dimensions $x$ by 100$/ x$
(a) Find any extreme values of $P$
(b) Give an interpretation in terms of perimeter of the rectangle for any values found in (a).

Linh Vu
Linh Vu
Numerade Educator
02:01

Problem 45

True or False If $f(c)$ is a local maximum of a continuous function $f$ on an open interval $(a, b),$ then $f^{\prime}(c)=0 .$ Justify your answer.

William Semus
William Semus
Numerade Educator
00:51

Problem 46

True or False If $m$ is a local minimum and $M$ is a local maximum of a continuous function $f$ on $(a, b),$ then $m<M .$ Justify your answer.

Linda Hand
Linda Hand
Numerade Educator
01:25

Problem 47

Multiple Choice Which of the following values is the absolute maximum of the function $f(x)=4 x-x^{2}+6$ on the interval $[0,4] ?$
(A) 0 $(\mathbf{B}) 2$ $(\mathbf{C}) 4$ $(\mathbf{D}) 6 \quad(\mathbf{E}) 10$

Linh Vu
Linh Vu
Numerade Educator
01:27

Problem 48

Multiple Choice If $f$ is a continuous, decreasing function on $[0,10]$ with a critical point at $(4,2),$ which of the following statements must be false?

(A) $f(10)$ is an absolute minimum of $f$ on $[0,10] .$
(B) $f(4)$ is neither a relative maximum nor a relative minimum.
(C) $f^{\prime}(4)$ does not exist.
(D) $f^{\prime}(4)=0$
(E) $f^{\prime}(4)<0$

Linda Hand
Linda Hand
Numerade Educator
01:24

Problem 49

Multiple Choice Which of the following functions has exactly two local extrema on its domain?
(A) $f(x)=|x-2|$
(B) $f(x)=x^{3}-6 x+5$
(C) $f(x)=x^{3}+6 x-5$
(D) $f(x)=\tan x$
(E) $f(x)=x+\ln x$

Linh Vu
Linh Vu
Numerade Educator
01:08

Problem 50

Multiple Choice If an even function $f$ with domain all real numbers has a local maximum at $x=a,$ then $f(-a)$
(A) is a local minimum.
(B) is a local maximum.
(C) is both a local minimum and a local maximum.
(D) could be either a local minimum or a local maximum.
(E) is neither a local minimum nor a local maximum.

Linda Hand
Linda Hand
Numerade Educator
03:20

Problem 51

Writing to Learn 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 (b) contradict the Extreme Value Theorem?
(d) Repeat parts (a) and (b) for $f(x)=(x-a)^{2 / 3}$ , replacing 2 by $a$ .

Norman Atentar
Norman Atentar
Numerade Educator
03:33

Problem 52

Writing to Learn Let $f(x)=\left|x^{3}-9 x\right|$
(a) Does $f^{\prime}(0)$ exist? (b) Does $f^{\prime}(3)$ exist?
(c) Does $f^{\prime}(-3)$ exist? (d) Determine all extrema of $f$

Norman Atentar
Norman Atentar
Numerade Educator
02:23

Problem 53

Cubic Functions Consider the cubic function
$$f(x)=a x^{3}+b x^{2}+c x+d$$
(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?

Linh Vu
Linh Vu
Numerade Educator
04:35

Problem 54

Proving Theorem 2 Assume that the function $f$ has a local maximum value at the interior point $c$ of its domain and that $f^{\prime}(c)$ exists.
(a) Show that there is an open interval containing $c$ such that $f(x)-f(c) \leq 0$ for all $x$ in the open interval.
(b) Writing to Learn Now explain why we may say
$$\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c} \leq 0$$
(c) Writing to Learn Now explain why we may say
$$\lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c} \geq 0$$
(d) Writing to Learn Explain how parts (b) and (c) allow us to conclude $f^{\prime}(c)=0 .$
(e) Writing to Learn Give a similar argument if $f$ has a local minimum value at an interior point.

Norman Atentar
Norman Atentar
Numerade Educator
01:42

Problem 55

Functions with $N o$ Extreme Values at Endpoints
(a) Graph the function
$$f(x)=\left\{\begin{array}{ll}{\sin \frac{1}{x},} & {x>0} \\ {0,} & {x=0}\end{array}\right.$$
Explain why $f(0)=0$ is not a local extreme value of $f$
(b) Group Activity Construct a function of your own that fails to have an extreme value at a domain endpoint.

Linh Vu
Linh Vu
Numerade Educator