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 : 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:

$x=0,\pm1$

x=1

There are no inflection points.

$x=\pm1$

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

Example Question #1 : Determine Points Of Inflection

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

Possible Answers:

(1,3)

(0,4)

(2,0)

(-0.9108,0)

$\left(\frac{1}{2},\frac{7}{2}\right)$

Correct answer:

$\left(\frac{1}{2},\frac{7}{2}\right)$

Explanation:

To find the x-coordinate of the point of inflection, we set the second derivative of the function equal to zero.

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

y'=6x^2-6x

y''=12x-6

0=12x-6

6=12x

$x=\frac{6}{12}=\frac{1}{2}$ .

To find the y-coordinate of the point, we plug the x-coordinate back into the original function.

$y=2\left(\frac{1}{2}\right)^3-3\left(\frac{1}{2}\right)^2+4$

$y=2\left(\frac{1}{8}\right)-3\left(\frac{1}{4}\right)+4$

$y=\frac{7}{2}$

The point is then $\left(\frac{1}{2},\frac{7}{2}\right)$ .

Report an Error

Example Question #11 : Determine Points Of Inflection

Determine the point(s) of inflection of $y=\frac{1}{x+4}$ .

Possible Answers:

(0,0)

(-4,0)

$\left(0,\frac{1}{4}\right)$

No points of inflection exist.

(-5,-1) and (-3,1)

Correct answer:

No points of inflection exist.

Explanation:

The points of inflection exist where the second derivative is zero.

$y=\frac{1}{x+4}$

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

$y''=\frac{2}{(x+4)^3}$ which can never be . Therefore, there are no points of inflection.

Report an Error

Example Question #12 : Determine Points Of Inflection

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

Possible Answers:

x=-3,3

No points of inflection exist.

x=0,3

x=3

x=-3

Correct answer:

No points of inflection exist.

Explanation:

The points of inflection will exist where the second derivative is zero.

$y=(x+3)^\frac{2}{3}$

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

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

$0=-\frac{2}{3}(x+3)^\frac{-4}{3}$

$0=-\frac{2}{3}\cdot\frac{1}{\sqrt[3]{x+3}^4}$ .

This will never be , so there are no points of inflection.

Report an Error

Example Question #13 : Determine Points Of Inflection

List the interval(s) where the function y=x^3-2x^2+2 is concave up.

Possible Answers:

$\left(0,\frac{3}{2}\right)$

The graph is never concave up.

The graph is always concave up.

$\left(-\infty,\frac{2}{3}\right)$

$\left(\frac{2}{3},\infty\right)$

Correct answer:

$\left(\frac{2}{3},\infty\right)$

Explanation:

The graph is concave up where the second derivative is positive. Let us first find out if there are any points of inflection to narrow our search.

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

y'=3x^2-4x

y''=6x-4

0=6x-4

$x=\frac{2}{3}$ .

Now we can perform the second derivative concavity test on points on either side of $x=\frac{2}{3}$ . Let us try x=0,1 .

The second derivative at x=0 gives us y''=-4 which is less than zero, so the graph is concave down in this interval. The second derivative at x=1 gives us y''=2 , which is positive. Hence, the graph is concave up on the interval $\left(\frac{2}{3},\infty\right)$ .

Report an Error

Example Question #13 : Determine Points Of Inflection

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

Possible Answers:

$x=\pm3$

x=0,3

There are no points of inflection.

$x=0,\sqrt{27}$

x=0

Correct answer:

There are no points of inflection.

Explanation:

The points of inflection will only exist where the second derivative is zero.

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

y'=2x

y''=2

Now $0\neq2$ therefore, there are no points of inflection.

Report an Error

Example Question #15 : Determine Points Of Inflection

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

Possible Answers:

(2,3)

(-2.28,-2.53),(0.92,3.92),(2.86,2.22)

There are no points of inflection.

$\left(-1,\frac{1}{4}\right)$

$\left(-1,\frac{1}{4}\right),(2,3)$

Correct answer:

$\left(-1,\frac{1}{4}\right),(2,3)$

Explanation:

The points of inflection, if any exist, will be found where the second derivative is zero.

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

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

y''=x^2-x-2

0=x^2-x-2

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

x=-1,2 .

To find the y-coordinates, we simply plug the x-coordinates in to the original function.

$y=\frac{1}{12}(-1)^4-\frac{1}{6}(-1)^3-(-1)^2+2(-1)+3=\frac{1}{4}$ .

So $(-1,\frac{1}{4})$ is an inflection point.

Also, $y=\frac{1}{12}(2)^4-\frac{1}{6}(2)^3-(2)^2+2(2)+3=3$ , so (2,3) is another inflection point.

Report an Error

Example Question #791 : Pre Calculus

List the intervals and determine where the graph of $y=\frac{1}{3}x^4+\frac{1}{3}x^3-6x^2+x+5$ is concave up and concave down.

Possible Answers:

The graph is concave up on $\left(-2,\frac{3}{2}\right)$ .

The graph is concave down on $(-\infty,-2)$ and $\left(\frac{3}{2},\infty\right)$ .

The graph is concave up on $(-\infty,0)$ .

The graph is concave down on $(0,\infty)$ .

The graph is always concave up.

The graph is concave up on $(-\infty,-2)$ and $\left(\frac{3}{2},\infty\right)$ .

The graph is concave down on $\left(-2,\frac{3}{2}\right)$ .

The graph is always concave down.

Correct answer:

The graph is concave up on $(-\infty,-2)$ and $\left(\frac{3}{2},\infty\right)$ .

The graph is concave down on $\left(-2,\frac{3}{2}\right)$ .

Explanation:

We need to set the second derivative equal to zero to determine where the inflection points are.

$y=\frac{1}{3}x^4+\frac{1}{3}x^3-6x^2+x+5$

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

y''=4x^2+2x-12

0=4x^2+2x-12

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

0=2x^2+x-6

0=(x+2)(2x-3)

$x=-2,\frac{3}{2}$ are the x-coordinates of our inflection points. Thus the intervals of concavity are $(-\infty,-2)$ , $\left(-2,\frac{3}{2}\right)$ , and $\left(\frac{3}{2},\infty\right)$ . We can use x=-3,0,2 as our test points.

y''(-3)=18>0 , so the graph is concave up on $(-\infty,-2)$ .

y''(0)=-12< 0 , so the graph is concave down on $\left(-2,\frac{3}{2}\right)$ .

y''(2)=8>0 , so the graph is concave up on $\left(\frac{3}{2},\infty\right)$ .

Report an Error

Example Question #792 : Pre Calculus

Determine the points of inflection, if any, of the following function:

f(x)=x^3-5x^2+2x+11

Possible Answers:

x=-7

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

$x=\frac{5}{3}$

x=4

$x=\frac{1}{4}$

Correct answer:

$x=\frac{5}{3}$

Explanation:

The points of inflection of a function are those at which its concavity changes. The concavity of a function is described by its second derivative, and when the second derivative is 0 a point of inflection occurs. We find the second derivative of the function and then set it equal to 0 to solve for the inflection points:

f(x)=x^3-5x^2+2x+11

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

f''(x)=6x-10

$6x-10=0\rightarrow 6x=10\rightarrow x=\frac{5}{3}$

So the function has only one point of inflection at x=5/3.

Report an Error

View Pre-Calculus Tutors

Tom
Certified Tutor

University of Wisconsin-Madison, Bachelor of Science, Atmospheric Sciences and Meteorology. Embry-Riddle Aeronautical Univers...

View Pre-Calculus Tutors

Satweka
Certified Tutor

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

View Pre-Calculus Tutors

Masoud
Certified Tutor

University of Guilan,, Bachelor of Science, Biological/Biosystems Engineering. University of Tehran, Master of Science, Anima...

All Precalculus Resources

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

Popular Subjects

Biology Tutors in Boston, Statistics Tutors in New York City, Reading Tutors in Dallas Fort Worth, MCAT Tutors in San Francisco-Bay Area, ACT Tutors in Miami, LSAT Tutors in Philadelphia, Biology Tutors in Los Angeles, ACT Tutors in Philadelphia, Chemistry Tutors in Miami, Calculus Tutors in Dallas Fort Worth

Popular Courses & Classes

MCAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Philadelphia, Spanish Courses & Classes in Houston, Spanish Courses & Classes in Boston, LSAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Boston, ACT Courses & Classes in Miami, ACT Courses & Classes in Seattle, ISEE Courses & Classes in Houston, SSAT Courses & Classes in Los Angeles

Popular Test Prep

ISEE Test Prep in Phoenix, ACT Test Prep in Seattle, GMAT Test Prep in Houston, LSAT Test Prep in Dallas Fort Worth, ACT Test Prep in Philadelphia, GRE Test Prep in Dallas Fort Worth, ISEE Test Prep in Atlanta, ISEE Test Prep in Houston, ACT Test Prep in New York City, SAT Test Prep in Seattle

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 : Determine Points Of Inflection

Example Question #2 : Determine Points Of Inflection

Example Question #1 : Determine Points Of Inflection

Example Question #11 : Determine Points Of Inflection

Example Question #12 : Determine Points Of Inflection

Example Question #13 : Determine Points Of Inflection

Example Question #13 : Determine Points Of Inflection

Example Question #15 : Determine Points Of Inflection

Example Question #791 : Pre Calculus

Example Question #792 : Pre Calculus

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: