Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

(a) Find the intervals of increase or decrease. …

04:17

Question

Answered step-by-step

Problem 39 Medium Difficulty

(a) Find the intervals of increase or decrease.
(b) Find the local maximum and minimum values.
(c) Find the intervals of concavity and the inflection points.
(d) Use the information from parts $ (a) - (c) $ to sketch the graph.
Check your work with a graphing device if you have one.

$ f(x) = \frac{1}{2} x^4 - 4x^2 + 3 $


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Madi Sousa
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Madi Sousa

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

01:43

Carson Merrill

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Related Topics

Derivatives

Differentiation

Volume

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Joseph Lentino

Boston College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

0:00


(a) Find the intervals…

05:23

(a) Find the intervals of …

06:33

(a) Find the intervals of …

0:00


(a) Find the intervals…

0:00


(a) Find the intervals…

06:08

(a) Find the intervals of …

02:43

(a) Find the intervals of …

07:59

(a) Find the intervals of …

0:00


(a) Find the intervals…

04:31

(a) Find the intervals of …

01:08

(a) Find the intervals of …

05:23

(a) Find the intervals of …

05:23

(a) Find the intervals of …

03:13

(a) Find the intervals of …

Watch More Solved Questions in Chapter 4

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
Problem 84
Problem 85
Problem 86
Problem 87
Problem 88
Problem 89
Problem 90
Problem 91
Problem 92
Problem 93
Problem 93

Video Transcript

and this problem, we are learning how derivatives affect the shape of a graph. More specifically, this problem is asking us how we can use the derivative to deduce important information about the function, like where the function is increasing or decreasing three intervals of con cavity. And it's critical numbers. And this is going to become important later on calculus when you start to learn curve sketching, so getting a great understanding of this now is going to be very helpful for you. Sorry. Pardon me for part A were given the function F of X equals in one half X to the fourth minus four X squared plus three. So we need to find the critical numbers and the intervals of con cavity. So the first thing that we're going to do is take the derivative F prime of X equals two X cubed minus eight x We could do some. Factoring will get two x times X minus two times X plus two. We're going to set that equal to zero to get the critical numbers. Then we'll solve for X so x zero X s two and excess negative too. So then what? We're going to Dio is we're going to create a table. You don't have to make a table, but just keep track of the intervals. We're going to make sure that we have the intervals using the critical numbers and then determine if our function would be increasing or decreasing in that interval. So the first interval is negative. Infinity too negative too. And our function is decreasing there. And if you don't know how I'm getting that, all you have to do is take a number within that interval, plug it into the function and see if you get a positive or a negative. For the second interval, we'd have negative 2 to 0 and we see our function is increasing there. Our third interval interval. Pardon me is from 0 to 2 and our function is decreasing there. And then finally, our last interval is to to infinity and our function is increasing there. So this is just another way to right. We found on the table at the vex increasing when X is an interval. Negative 20 and two comma infinity and f of X is decreasing on the interval when X is in thes two intervals negative and they needed to negative negative two or two or pardon me, you 02 and then for party were told. What are the maxes and men's of this function? What we have the critical numbers. All we have to do now is plugged them back into our original function. So f of negative two is negative. Five f of two was negative. Five and F of zero is three. So clearly we have a minimum, a negative two and two and a maximum at zero for part C, we're told. Let's check the con cavity. So when you hear the word con cavity your mind, you immediately go to second derivative. We need the second derivative test to determine the intervals of con Cavity. So F double prime of X is six x squared minus eight. And now this one's a little bit more confusing than before. We have to do some clever factoring. So this is the same thing as saying six times X squared, minus 4/3. And then we want critical numbers, Remember? So we have to factor this further. We're going to get six times X minus two over the square root of three times X plus two over the square root of three. So then we have our critical numbers so we can determine con cavity. Our function is con cave up on the interval. Negative infinity negative to over screwed of three and two over. Screwed of three common infinity And then we see a con cave down behavior on the interval negative to over the square root of 3 to 2 over the square root of three. So what does this mean? We change from going from concave up to con cave down So we see an inflection point when x is negative two over the skirt of three and when X is two over the square root of three. And now finally the problem asks us, Well, what's going to happen with the graph? Can we check our information from the graph? This is the graph of the function and you can clearly see that our information that we found using just differential calculus matches the function. We see our minimums where we found them, our maximum where we found them. Our con cavity matches as well. So I hope this problem helped to understand a little bit more about how the derivative, um, affect the shape of the graph more specifically, how we can use the derivative to deduce important information from a function such as the intervals of increase in decrease and it's common cavity.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
126
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
62
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Volume

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Joseph Lentino

Boston College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

04:35

Volume - Intro

In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.

Video Thumbnail

06:14

Review

A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.

Join Course
Recommended Videos

0:00

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

05:23

(a) Find the intervals of increase or decrease. (b) Find the local maximum and …

06:33

(a) Find the intervals of increase or decrease. (b) Find the local maximum and …

0:00

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

0:00

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

06:08

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

02:43

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

07:59

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

0:00

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

04:31

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

01:08

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…

05:23

(a) Find the intervals of increase or decrease. (b) Find the local maximum and …

05:23

(a) Find the intervals of increase or decrease. (b) Find the local maximum and …

03:13

(a) Find the intervals of increase or decrease. (b) Find the local maximum and…
Additional Mathematics Questions

01:35

Junior put 10 pesos in his piggy bank yesterday, May 30,2014 of he is planni…

00:56

find the product of the quotient of -132 and 12, and the difference of -105 …

02:02

A pile of blocks has 60 blocks in the bottom row, 54 block in the second row…

00:49

What must be the value of n so that 2n is equal to 1/2?

00:26

What is the constant of proportionality in the equation y=2.5xy=2.5xy, equal…

00:19

8. Assuming they work at the same rate, howlong (S) will it take 2 housekeep…

01:29

The four children in the Rivera family are Reynaldo, Ramiro, Shakira, and Sa…

01:16

The profit made by a manufacturing company producing bags is givenby the fun…

01:39

A cylinder is 154 inches high and 3 inches in radius, find its volume.
…

02:52

a1=2, a3=8, n=4, find d and the 4th term

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started