Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Introductory Calculus

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 #1 : Find The Critical Numbers Of A Function

Find the critical number(s) of the function

f(x)=x^2 .

Possible Answers:

0.5

DNE

Correct answer:

Explanation:

To find the critical numbers, find the values for x where the first derivative is 0 or undefined.

For the function

f(x)=ax^n

The first derivative is

$f^{'}(x)=anx^{n-1}$

So for

f(x)=x^2

the first derivative is

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

Setting that equal to zero

2x=0

We find

x=0

Report an Error

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical number(s) of the function

f(x)=x^3-3x^2 .

Possible Answers:

DNE

0, 2

Correct answer:

0, 2

Explanation:

To find the critical numbers, find the values for x where the first derivative is 0 or undefined.

For the function

f(x)=ax^n .

The first derivative is

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

So for

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

the first derivative is

$f'(x)=3x^{3-1}-3(2)x^{2-1}=3x^2-6x=3x(x-2)$ .

Settig the first derivative equal to zero

f'(x)=3x(x-2) = 0

yields the values

x=0,2

Report an Error

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

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

Possible Answers:

x=-6,x=2

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

x=-2,x=6

x=5,x=7

x=-7,x=-5

Correct answer:

x=-2,x=6

Explanation:

The critical numbers of a function are those at which its first derivative is equal to 0. These points tell where the slope of the function is 0, which lets us know where the minimums and maximums of the function are. First we find the derivative of the function, then we set it equal to 0 and solve for the critical numbers:

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

f'(x)=x^2-4x-12=(x-6)(x+2)=0

x=-2,x=6

Report an Error

Example Question #7 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

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

Possible Answers:

x=-3,x=-2

x=2,x=3

x=5,x=12

x=-12,x=5

$x=-\frac{2}{3},x=4$

Correct answer:

x=2,x=3

Explanation:

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

f'(x)=2x^2-10x+12=2(x^2-5x+6)=2(x-2)(x-3)=0

x=2,x=3

Report an Error

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

$f(x)=x^3+\frac{3}{2}x^2-90x$

Possible Answers:

x=-3,x=4

x=-5,x=6

x=-4,x=3

$x=-\frac{3}{4},x=1$

x=-6,x=5

Correct answer:

x=-6,x=5

Explanation:

$f(x)=x^3+\frac{3}{2}x^2-90x$

f'(x)=3x^2+3x-90=3(x^2+x-30)=3(x+6)(x-5)=0

x=-6,x=5

Report an Error

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function $f(x)=3x^{3}$ .

Possible Answers:

x=0,3

x=0

x=0,-3

Answer not listed

x=3,-3

Correct answer:

x=0

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

$f(x)=3x^{3}$

$f'(x)=9x^{2}$

Setting f'(x)=0 :

$9x^{2}=0$

$x^{2}=0$

x=0

Report an Error

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function $f(x)=x^{2}-2x+1$ .

Possible Answers:

Correct answer not listed

x=1

x=1,0

x=1,-1

x=-1

Correct answer:

x=1

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

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

f'(x)=2x-2

Setting f'(x)=0 :

2x-2=0

2x=2

x=1

Report an Error

Example Question #101 : Derivatives

Find the critical value(s) of a function $f(x)=(x+1)^{3}$ .

Possible Answers:

x=-1

x=0,1

x=1,-1

x=0,-1

x=1

Correct answer:

x=-1

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

$f(x)=(x+1)^{3}$

$f'(x)=3(x+1)^{2}$

Setting f'(x)=0 :

$3(x+1)^{2}=0$

$(x+1)^{2}=0$

x+1=0

x=-1

Report an Error

Example Question #101 : Derivatives

Find the critical numbers of the following function:

f(x)=x^3-5x^2+4x+9

Possible Answers:

$x=3-\sqrt{6},x=3+\sqrt{6}$

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

x=-3, x=4

$x=\frac{1}{2}(7-3\sqrt{5}),x=\frac{1}{2}(7+3\sqrt{5})$

$x=10-\sqrt{11},x=10+\sqrt{11}$

Correct answer:

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

Explanation:

The critical numbers of a function are the points at which its slope is zero, which tells where the function has a minimum or maximum. The slope of a function is described by its derivative, so we'll take the derivative of the function and set it equal to 0 to solve for the x values of the critical numbers:

f(x)=x^3-5x^2+4x+9

f'(x)=3x^2-10x+4

It appears the derivative cannot be factored to solve for x, so we'll have to use the quadratic formula to find the critical numbers:

3x^2-10x+4=0

$\frac{10\pm \sqrt{(-10)^2-4(3)(4)}}{2(3)}=\frac{5}{3}\pm\frac{1}{3}\sqrt{13}=\frac{1}{3}(5\pm \sqrt{13})$

So the critical numbers occur at the following two values of x:

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

Report an Error

Example Question #771 : Pre Calculus

Given the following function, find the critical numbers: y= 3x^2-2x-50

Possible Answers:

$-\frac{2}{3}$

$-\frac{1}{3}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

Critical numbers are where the slope of the function is equal to zero or undefined.

Find the derivative and set the derivative function to zero.

y= 3x^2-2x-50

y'= 6x-2 = 0

6x=2

$x=\frac{1}{3}$

There is only one critical value at $x=\frac{1}{3}$ .

Report an Error

View Pre-Calculus Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Pre-Calculus Tutors

Abhisek
Certified Tutor

University of Florida, Bachelor of Engineering, Computer Science.

View Pre-Calculus Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

All Precalculus Resources

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

Popular Subjects

Spanish Tutors in New York City, Statistics Tutors in Boston, MCAT Tutors in Atlanta, Biology Tutors in Denver, ISEE Tutors in San Diego, Statistics Tutors in New York City, Statistics Tutors in Philadelphia, Math Tutors in Philadelphia, ACT Tutors in San Francisco-Bay Area, GRE Tutors in San Diego

Popular Courses & Classes

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

Popular Test Prep

ISEE Test Prep in Boston, SSAT Test Prep in Houston, GMAT Test Prep in Los Angeles, ISEE Test Prep in Chicago, GRE Test Prep in New York City, ACT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Philadelphia, SSAT Test Prep in Philadelphia, ISEE Test Prep in Miami, MCAT Test Prep in San Francisco-Bay Area

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 : Introductory Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #7 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #101 : Derivatives

Example Question #101 : Derivatives

Example Question #771 : Pre Calculus

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: