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(a) Investigate the family of polynomials given b…

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Problem 38 Medium Difficulty

Investigate the family of curves given by the equation $ f(x) = x^4 + cx^2 + x $. Start by determining the transitional value of $ c $ at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. There is another transitional value of $ c $ at which the number of critical numbers changes. Try to discover it graphically. Then prove what you have discovered.


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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

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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.

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Related Topics

Derivatives

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Volume

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Samuel Hannah

University of Nottingham

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.

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