We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus BC : Inflection Points

Study concepts, example questions & explanations for AP Calculus BC

Create An Account

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #25 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=2x^{3} - 3x^{2} - 2x^{4} - 8x^{5} - 1\end{align*}$

Possible Answers:

-1.424

-0.475

-3.322

$\text{There are no points of inflection.}$

Correct answer:

-0.475

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&6x^{2} - 6x - 8x^{3} - 40x^{4}\\&\text{And our second is:}\\&12x - 24x^{2} - 160x^{3} - 6\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.475,0.162-0.230i,0.162+0.230i\\&\text{We find one real unique root at the point }x=-0.475\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #26 : Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=3x + 8x^{2} - 3x^{3} - 10x^{4} - 5\end{align*}$

Possible Answers:

-0.448,0.298

$\text{There are no points of inflection.}$

-0.896,1.191

-3.134,2.084

Correct answer:

-0.448,0.298

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x - 9x^{2} - 40x^{3} + 3\\&\text{And our second is:}\\&16 - 120x^{2} - 18x\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.448,0.298\\&\text{We find real unique roots at the points }x=-0.448,0.298\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #26 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=\frac{1}{2^{(3x)}}- x - 10x^{2} + x^{3} - 10x^{4} + 6x^{5} + 3\end{align*}$

Possible Answers:

2.182

7.637

$\text{There are no points of inflection.}$

1.091

Correct answer:

1.091

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3x^{2} -\frac{ (3ln(2))}{2^{(3x)}}- 20x - 40x^{3} + 30x^{4} - 1\\&\text{And our second is:}\\&6x +\frac{ (9ln(2)^{2})}{2^{(3x)}}- 120x^{2} + 120x^{3} - 20\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=1.091\\&\text{We find one real unique root at the point }x=1.091\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #28 : Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&3^{(3x)} - 4x + 10x^{3} + 10x^{4} + 2x^{5}\end{align*}$

Possible Answers:

-0.026

-0.154

-0.077

$\text{There are no points of inflection.}$

Correct answer:

-0.154

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&3\cdot 3^{(3x)}ln(3) + 30x^{2} + 40x^{3} + 10x^{4} - 4\\&\text{And our second is:}\\&60x + 9\cdot 3^{(3x)}ln(3)^{2} + 120x^{2} + 40x^{3}\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x yields:}\\&x=-0.154\\&\text{We find one real unique root at the point }x=-0.154\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #29 : Derivatives

$\begin{align*}&\text{Find any points of inflection for the function:}\\&e^{(2x)} - 8x + 8x^{2} + 5x^{3} - 2x^{4} - 7x^{5} - 2\end{align*}$

Possible Answers:

0.132

$\text{There are no points of inflection.}$

0.660

0.330

Correct answer:

0.660

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative, so let's do it in steps. For the}\\&\text{function we're given, our first derivative is :}\\&16x + 2e^{(2x)} + 15x^{2} - 8x^{3} - 35x^{4} - 8\\&\text{And our second is:}\\&30x + 4e^{(2x)} - 24x^{2} - 140x^{3} + 16\\&\text{For a reminder of derivative rules:}\\&d[e^{ax}]=ae^{ax}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=0.660\\&\text{We find one real unique root at the point }x=0.660\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #26 : Inflection Points

$\begin{align*}&\text{Find any points of inflection for the function:}\\&4x + ln(5x) - 6x^{2} - 10x^{3} + 6x^{4} - 5\end{align*}$

Possible Answers:

-0.292,1.012

$\text{There are no points of inflection.}$

-1.462,5.058

-0.877,0.506

Correct answer:

-0.292,1.012

Explanation:

$\begin{align*}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.292,1.012,0.057-0.209i,0.057+0.209i\\&\text{We find real unique roots at the points }x=-0.292,1.012\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #31 : Second Derivatives

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=ln(5x) + 8x^{2} - 10x^{3} + 5x^{4} + 3\end{align*}$

Possible Answers:

-1.111,4.555

-0.062,1.301

$\text{There are no points of inflection.}$

-0.185,0.651

Correct answer:

-0.185,0.651

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative. For the}\\&\text{function we're given, our first derivative is :}\\&16x +\frac{ 1}{x}- 30x^{2} + 20x^{3}\\&\text{And our second is:}\\&60x^{2} -\frac{ 1}{x^{2}}- 60x + 16\\&\text{For a reminder of derivative rules:}\\&d[ln(ax)]=\frac{dx}{x}\\&d[x^a]=ax^{a-1}dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.185,0.651,0.267-0.259i,0.267+0.259i\\&\text{We find real unique roots at the points }x=-0.185,0.651\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at every point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #301 : Ap Calculus Bc

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=2^{(6x)} - 6x - 5x^{2} + 3x^{3} + 9\end{align*}$

Possible Answers:

-0.013

-0.024

-0.094

$\text{There are no points of inflection.}$

Correct answer:

-0.094

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative. For the function we're given, our}\\&\text{first derivative is :}\\&6\cdot 2^{(6x)}ln(2) - 10x + 9x^{2} - 6\\&\text{And our second is:}\\&18x + 36\cdot 2^{(6x)}ln(2)^{2} - 10\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.094\\&\text{We find one real unique root at the point }x=-0.094\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

Example Question #302 : Ap Calculus Bc

$\begin{align*}&\text{Calculate any and all points of inflection for: }\\&f(x)=10x + 2^{(6x)} + 2x^{2} - 2x^{3}\end{align*}$

Possible Answers:

-0.018

-0.160

-0.040

$\text{There are no points of inflection.}$

Correct answer:

$\text{There are no points of inflection.}$

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative. For the function we're given, our}\\&\text{first derivative is :}\\&4x + 6\cdot 2^{(6x)}ln(2) - 6x^{2} + 10\\&\text{And our second is:}\\&36\cdot 2^{(6x)}ln(2)^{2} - 12x + 4\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=-0.160-0.553i\\&\text{Since we find no real roots, this function has no points of inflection.}\\&\text{In this case, the function is always concave up.}\\&\end{align*}$

Report an Error

Example Question #303 : Ap Calculus Bc

$\begin{align*}&\text{Find any points of inflection for the function:}\\&2^{(3x)} - 6x + 2x^{2} - 7x^{3} - 9x^{5} + 1\end{align*}$

Possible Answers:

0.214

1.070

0.642

$\text{There are no points of inflection.}$

Correct answer:

0.214

Explanation:

$\begin{align*}&\text{A point of inflection of a function is where its concavity changes}\\&\text{sign. To find the concavity of a function, we'll wish to first}\\&\text{find its second derivative. For the function we're given, our}\\&\text{first derivative is :}\\&4x + 3\cdot 2^{(3x)}ln(2) - 21x^{2} - 45x^{4} - 6\\&\text{And our second is:}\\&9\cdot 2^{(3x)}ln(2)^{2} - 42x - 180x^{3} + 4\\&\text{For a reminder of derivative rules:}\\&d[b^{ax}]=ab^{ax}ln(b)dx\\&\text{Now, if a point of inflection exists, it's where this function equals zero with real roots.}\\&\text{Solving for x with a calculator yields:}\\&x=0.214\\&\text{We find one real unique root at the point }x=0.214\\&\text{Now to determine if a point is one of inflection, plug in values on either side of it,}\\&\text{and see if the sign changes. Say }\pm0.005\text{ its value.}\\&\text{With this check we find that at this point the concavity of the function changes.}\end{align*}$

Report an Error

View AP Calculus BC Tutors

James
Certified Tutor

University of Georgia, Bachelor in Arts, Political Science and Government.

View AP Calculus BC Tutors

Leo
Certified Tutor

Columbia International University, Bachelor of Science, Chemistry.

View AP Calculus BC Tutors

Reza
Certified Tutor

University of Sherbrooke, Doctor of Philosophy, Mathematics. University of Manitoba, Master of Science, Mathematics.

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Calculus Tutors in San Diego, Calculus Tutors in Dallas Fort Worth, Computer Science Tutors in Boston, GMAT Tutors in Atlanta, English Tutors in New York City, Biology Tutors in New York City, English Tutors in Atlanta, LSAT Tutors in Los Angeles, ISEE Tutors in San Francisco-Bay Area, Computer Science Tutors in New York City

Popular Courses & Classes

ISEE Courses & Classes in Boston, GRE Courses & Classes in Boston, SSAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Boston, GMAT Courses & Classes in Houston, Spanish Courses & Classes in Chicago, SAT Courses & Classes in Denver, LSAT Courses & Classes in Atlanta, GRE Courses & Classes in Philadelphia, Spanish Courses & Classes in Seattle

Popular Test Prep

MCAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Atlanta, SSAT Test Prep in Denver, SAT Test Prep in Philadelphia, LSAT Test Prep in Washington DC, ACT Test Prep in Atlanta, ISEE Test Prep in Washington DC, ACT Test Prep in New York City, GMAT Test Prep in Houston, GRE Test Prep in New York City

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

AP Calculus BC : Inflection Points

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #25 : Second Derivatives

Example Question #26 : Derivatives

Example Question #26 : Second Derivatives

Example Question #28 : Derivatives

Example Question #29 : Derivatives

Example Question #26 : Inflection Points

Example Question #31 : Second Derivatives

Example Question #301 : Ap Calculus Bc

Example Question #302 : Ap Calculus Bc

Example Question #303 : Ap Calculus Bc

All AP Calculus BC Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: