We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #111 : Derivatives

Find the critical values of the following function.

f(x)=x^2-6x+3

Possible Answers:

Correct answer:

Explanation:

To solve, simply differentiate using the power rule, as outlined below.

Power rule states,

$(x^n)'=nx^{n-1}$ .

Thus given,

f(x)=x^2-6x+3

our first derivative is:

$f'(x)=2x^{2-1}-6x^{1-1}=2x-6$

Then plug in 0 for f(x) to find when our function is equal to 0.

Thus,

0=2x-6

2x=6

x=3

Report an Error

Example Question #1 : Determine Points Of Inflection

Determine the location of the points of inflection for the following function:

$f(x)=\frac{1}{12}x^4-\frac{1}{2}x^3-5x^2+4x+17$

Possible Answers:

x=4,x=7

x=-5,x=2

x=-2,x=5

x=-4,x=3

x=-3,x=4

Correct answer:

x=-2,x=5

Explanation:

The points of inflection of a function are the points at which its concavity changes. The concavity of a function is described by its second derivative, which will be equal to zero at the inflection points, so we'll start by finding the first derivative of the function:

$f(x)=\frac{1}{12}x^4-\frac{1}{2}x^3-5x^2+4x+17$

$f'(x)=\frac{1}{3}x^3-\frac{3}{2}x^2-10x+4$

Next we'll take the derivative one more time to get the second derivative of the original function:

$f'(x)=\frac{1}{3}x^3-\frac{3}{2}x^2-10x+4$

f''(x)=x^2-3x-10

Now that we have the second derivative of the function, we can set it equal to 0 and solve for the points of inflection:

$x^2-3x-10=0\rightarrow (x-5)(x+2)=0\rightarrow x=-2,x=5$

Report an Error

Example Question #2 : Determine Points Of Inflection

Find the points of inflection of the following function:

$f(x)=\frac{1}{4}x^4-5x^3+\frac{63}{2}x^2-21$

Possible Answers:

x=-3,x=5

x=2,x=4

x=-1,x=5

x=3,x=7

x=-11,x=-7

Correct answer:

x=3,x=7

Explanation:

The points of inflection of a function are those at which its second derivative is equal to 0. First we find the second derivative of the function, then we set it equal to 0 and solve for the inflection points:

$f(x)=\frac{1}{4}x^4-5x^3+\frac{63}{2}x^2-21$

f'(x)=x^3-15x^2+63x

f''(x)=3x^2-30x+63=3(x^2-10x+21)=3(x-3)(x-7)=0

x=3,x=7

Report an Error

Example Question #1 : Determine Points Of Inflection

Find the inflection points of the following function:

$f(x)=\frac{1}{6}x^4-x^3-28x^2+9$

Possible Answers:

x=-7,x=4

x=-4,x=7

$x=-3,x=-\frac{2}{3}$

x=-5,x=3

x=-3,x=5

Correct answer:

x=-4,x=7

Explanation:

$f(x)=\frac{1}{6}x^4-x^3-28x^2+9$

$f'(x)=\frac{2}{3}x^3-3x^2-56x$

f''(x)=2x^2-6x-56=2(x^2-3x-28)=2(x-7)(x+4)=0

x=-4,x=7

Report an Error

Example Question #1 : Determine Points Of Inflection

Determine the points of inflection of the following function:

$f(x)=\frac{1}{12}x^4+\frac{1}{3}x^3-\frac{15}{2}x^2+17$

Possible Answers:

$x=-\frac{1}{3},x=2$

x=-2,x=4

x=-3,x=5

x=-5,x=3

x=-4,x=2

Correct answer:

x=-5,x=3

Explanation:

$f(x)=\frac{1}{12}x^4+\frac{1}{3}x^3-\frac{15}{2}x^2+17$

$f'(x)=\frac{1}{3}x^3+x^2-15x$

f''(x)=x^2+2x-15=(x+5)(x-3)=0

x=-5,x=3

Report an Error

Example Question #116 : Derivatives

Determine the x-coordinate of the inflection point of the function y=x^3-4x^2+3 .

Possible Answers:

$-\frac{4}{3}$

$\frac{8}{3}$

$\frac{4}{3}$

Correct answer:

$\frac{4}{3}$

Explanation:

The point of inflection exists where the second derivative is zero.

y=x^3-4x^2+3

y'=3x^2-8x

y''=6x-8 , and we set this equal to zero.

0=6x-8

8=6x

$x=\frac{8}{6}=\frac{4}{3}$

Report an Error

Example Question #117 : Derivatives

Find the x-coordinates of all points of inflection of the function y=x^4+2x^3-3x^2+1 .

Possible Answers:

$0,-\frac{3}{4}\pm\frac{\sqrt{33}}{4}$

2,-3

$-\frac{1}{3}\pm\frac{\sqrt{2}}{3}$

$-\frac{1}{2}\pm\frac{\sqrt{3}}{2}$

There are no points of inflection

Correct answer:

$-\frac{1}{2}\pm\frac{\sqrt{3}}{2}$

Explanation:

We set the second derivative of the function equal to zero to find the x-coordinates of any points of inflection.

y=x^4+2x^3-3x^2+1

y'=4x^3+6x^2-6x

y''=12x^2+12x-6

0=12x^2+12x-6

0=6(2x^2+2x-1)

$0\neq 6$

0=2x^2+2x-1 , and the quadratic formula yields

$x=-\frac{1}{2}\pm\frac{\sqrt{3}}{2}$ .

Report an Error

Example Question #118 : Derivatives

Determine the x-coordinate(s) of the point(s) of inflection of the function $y=\frac{1}{6}x^4-\frac{2}{3}x^3-15x^2+12x+1$ .

Possible Answers:

x=-5.6,8.2

x=0

There are no points of inflection.

x=-3

x=-3,5

Correct answer:

x=-3,5

Explanation:

Any points of inflection that exist will be found where the second derivative is equal to zero.

$y=\frac{1}{6}x^4-\frac{2}{3}x^3-15x^2+12x+1$

$y'=\frac{2}{3}x^3-2x^2-30x+12$

y''=2x^2-4x-30

0=2x^2-4x-30

0=2(x^2-2x-15) .

Since $0\neq2$ , we can focus on 0=x^2-2x-15 . Thus

0=(x+3)(x-5) , and x=-3,5 .

Report an Error

Example Question #1 : Determine Points Of Inflection

Find the x-coordinate(s) of the point(s) of inflection of $y=\frac{1}{6}x^4-x^2-3x+1$ .

Possible Answers:

There are no inflection points.

$x=0,\pm1$

x=-1

$x=\pm1$

x=1

Correct answer:

$x=\pm1$

Explanation:

The inflection points, if they exist, will occur where the second derivative is zero.

$y=\frac{1}{6}x^4-x^2-3x+1$

$y'=\frac{2}{3}x^3-2x-3$

y''=2x^2-2

0=2x^2-2

0=2(x^2-1)

0=x^2-1

$x=\pm1$

Report an Error

Example Question #2 : Determine Points Of Inflection

Find the point(s) of inflection of the function y=4x^3-3x+2 .

Possible Answers:

x=0

There is no point of inflection.

(2,28)

(0,2)

x=2

Correct answer:

(0,2)

Explanation:

The point of inflection will exist where the second derivative equals zero.

y=4x^3-3x+2

y'=12x^2-3

y''=24x

0=24x

x=0 .

Now we need the y-coordinate of the point.

y=4(0)^3-3(0)+2=2

Thus the inflection point is at (0,2) .

Report an Error

View Pre-Calculus Tutors

Chernyeh
Certified Tutor

Purdue University-Main Campus, Bachelor of Science, Business Administration and Management.

View Pre-Calculus Tutors

Bhupinder
Certified Tutor

Panjab University, Bachelor of Engineering, Biotechnology. Hamburg University of Technology, Master of Science, Chemical Engi...

View Pre-Calculus Tutors

Nicole
Certified Tutor

College of St Benedict, Bachelor in Arts, Economics. University of South Florida-Main Campus, Master of Arts, Economics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in New York City, Physics Tutors in Philadelphia, French Tutors in Philadelphia, French Tutors in San Francisco-Bay Area, GMAT Tutors in New York City, French Tutors in San Diego, SAT Tutors in Atlanta, Physics Tutors in Seattle, GRE Tutors in Seattle, Algebra Tutors in San Diego

Popular Courses & Classes

MCAT Courses & Classes in Los Angeles, GRE Courses & Classes in Boston, GRE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Boston, GRE Courses & Classes in Chicago, ISEE Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Houston, GRE Courses & Classes in Atlanta

Popular Test Prep

SAT Test Prep in Philadelphia, GMAT Test Prep in Phoenix, GRE Test Prep in Philadelphia, MCAT Test Prep in Washington DC, ACT Test Prep in San Francisco-Bay Area, ACT Test Prep in Seattle, SSAT Test Prep in Boston, ISEE Test Prep in Seattle, GMAT Test Prep in San Diego, ISEE Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #111 : Derivatives

Example Question #1 : Determine Points Of Inflection

Example Question #2 : Determine Points Of Inflection

Example Question #1 : Determine Points Of Inflection

Example Question #1 : Determine Points Of Inflection

Example Question #116 : Derivatives

Example Question #117 : Derivatives

Example Question #118 : Derivatives

Example Question #1 : Determine Points Of Inflection

Example Question #2 : Determine Points Of Inflection

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: