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

Question

Answered step-by-step

Problem 40 Hard Difficulty

(a) Investigate the family of polynomials given by the equation $ f(x) = 2x^3 + cx^2 + 2x $. For what values of $ c $ does the curve have maximum and minimum points?
(b) Show that the minimum and maximum points of every curve in the family lie on the curve $ y = x - x^3 $. Illustrate by graphing this curve and several members of the family.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Bobby Barnes
University of North Texas

Like

Report

Textbook Answer

Official textbook answer

Video by Bobby Barnes

Numerade Educator

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

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 6

Graphing with Calculus and Calculators

Related Topics

Derivatives

Differentiation

Volume

Discussion

You must be signed in to discuss.
Top Calculus 2 / BC Educators
Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

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

07:44

(a) Investigate the family…

01:25

(a) Investigate the family…

02:32

(a) Investigate the family…

0:00

The figure shows graphs (i…

01:21

Investigate the family of …

01:12

Investigate the family of …

04:18

Investigate the family 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

Video Transcript

we want to investigate family. A polynomial is given by the equation of accident to two x cubed plus C x squared plus two X, and we want to find for what values of seed does the curve have a maximum and minimum points. And then we want to show that these minimum the maximum points lie on the curve of why is equal to X minus excuse. All right, so first, let's go ahead and figure out what values of sea will give them maximum. So person, when you do, is take the derivative of every Becks and then find our critical values by saying the derivative equal deserve So taking that derivative there, we're going to get that f prime of X is equal to Well, we'll use powerful to take those derivatives. So we get six at squared plus two c x lost to physical zero, and we could go ahead and just divide all this by two. And I would give us three x squared. What's C. X plus one is a good zero, and now we can go ahead and used the quadratic equation too solved for this and just trying that out is going to be, and you actually go ahead and put lives above it. So three is a see is B and I should put quotation there on that. And one is C with quotations around that. So I'm just gonna go ahead and write out with standard quote. Our equation is so negative B plus or minus east where minus for a c all over two a and then plug it. Everything in we'll get negative C plus or minus the square root of C squared minus Well, a is three c is one, so we'll get 12 all over and then two times a is six. So we get this here for all of our possible solutions. And so the thing we're going to need to make sure is that this radical here is defied. So that's going to happen when let me go. Let me just move this up here when C squared minus 12 is strictly greater than zero. So I can add 12 over I get see greater than or equal to 12 c squared greater than a good 12 and then we can go ahead and take the square root. So I was gonna say CIA strictly greater than the square root of 12. Well, that's one solution. But if C is negative, we can also have that they're a swell And then we could just multiply that negative over and we get see lesson Every go to the negative square root. So these two here will be the values of See that will give. I'll say maximum Berman. So let me just go ahead and write down here. So I see is in the interval of negative infinity two negative 12 Union 12 to Infinity than this here will give us a maximum or a minimum for that class of functions. All right, now, the next thing we want to do is to show that thes maximums. So remember, the maximum men that we have is really this value over here. So we want to show really that X equal to negative C plus or minus this word of C squared minus 12 over six. Uh, you go ahead and make that Maur like negative Chlo. So, Myers 12 we want to show that this gives the same output or our function f of X as well as this line y minus X cubed. So Let's just set these two functions equal to each other and see what we get. So I'm gonna set f of X equal to why so F of X is two x cubed plus see X squared plus two x and why was X minus X cute social? See where these two lines would intersect so I can subtract X and add Execute Gilbert, which would give three x cute plus c X squared plus X is equal to zero. Let's go ahead and factor out an X So we get X times three X squared plus c X plus one is equal to zero. So this here tells us by the zero product property either X is equal to zero War three x squared plus c X plus one is equal to zero. And you might know this, that we've already solved for this because when we looked at the derivative over here and we said equal zero and we divided it by two, we ended up with that exact same what drug? So the solutions, forearm axes and men's will all be the same, so you could go ahead and solve this and then, um, do it that way. But I think this is good place to just go ahead and stop

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
71
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
49
Hosted by: Alonso M
See More

Related Topics

Derivatives

Differentiation

Volume

Top Calculus 2 / BC Educators
Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Samuel Hannah

University of Nottingham

Michael Jacobsen

Idaho State University

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

07:44

(a) Investigate the family of polynomials given by the equation $ f(x) = cx^4 -…

01:25

(a) Investigate the family of polynomials given by the equation $f(x)=c x^{4}-2…

02:32

(a) Investigate the family of polynomials given by the equation $f(x)=2 x^{3}+…

0:00

The figure shows graphs (in blue) of several members of the family of polynomia…

01:21

Investigate the family of curves $f(x)=e^{x}-c x .$ In particular, find the lim…

01:12

Investigate the family of curves $ f(x) = e^x - cx $. In particular, find the l…

04:18

Investigate the family of curves given by $ f(x) = xe^{-cx} $, where $ c $ is a…
Additional Mathematics Questions

02:47

Given the graph of f (x)
a) Sketch a possible graph of y = f (x) on the s…

02:54

Given the Lorenz curve
L=2.52' 3.39x3 + 2.0922 0.25
What percent …

04:25

point) The graph of f is shown below. Evaluate each integral by interpreting…

02:34

The expressions of the electronic Hamiltonian of the H2 ion in the Is H-atom…

03:12

Use the appropriate formula to find the value of the annuity: b. Find the in…

03:55

10. Plot apogee radius and eccentricity vs. Perigee radius of sun-synchronou…

02:25

A consumer has $ 240 to spend on two commodities, the first of which costs $…

01:28

Evaluate MPk and MP_ for the production function Q =2LK + VL given that the …

11:06

8. In each figure below, imagine drawing the diagonals AC and BD Find the mi…

02:41

large tank contains 120 litres of water in which 18 grams of salt is dissolv…

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
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started