We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Decreasing Intervals

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Intervals

Find the interval(s) where the following function is decreasing. Graph to double check your answer.

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

Possible Answers:

Always

$(-\infty,-5) \cup (1,\infty)$

(-5,1)

Never

Correct answer:

(-5,1)

Explanation:

To find when a function is decreasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is negative.

First, take the derivative:

y'=x^2+4x-5

Set equal to 0 and solve:

x^2+4x-5=0

(x+5)(x-1)=0

x=-5,1

Now test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of -6, 0, and 2.

y'=x^2+4x-5

y'(-6)=(-6)^2+4(-6)-5=7

y'(0)=(0)^2+4(0)-5=-5

y'(2)=(2)^2+4(2)-5=7

Since the only value that is negative is when x=0, the interval is only decreasing on the interval that includes 0. Therefore, our answer is:

(-5,1)

Report an Error

Example Question #2 : Intervals

Find the interval(s) where the following function is decreasing. Graph to double check your answer.

$y=\frac{1}{3}x^3-5x^2+9x+5$

Possible Answers:

Always

$(-\infty,1)\cup (9,\infty)$

Never

(1,9)

Correct answer:

(1,9)

Explanation:

To find when a function is decreasing, you must first take the derivative, then set it equal to 0, and then find between which zero values the function is negative.

First, take the derivative:

y'=x^2-10x+9

Set equal to 0 and solve:

x^2-10x+9=0

(x-9)(x-1)=0

x=1,9

Now test values on all sides of these to find when the function is negative, and therefore decreasing. I will test the values of 0, 2, and 10.

y'=x^2-10x+9

y'(0)=(0)^2-10(0)+9=9

y'(2)=(2)^2-10(2)+9=-7

y'(10)=(10)^2-10(10)+9=9

Since the only value that is negative is when x=0, the interval is only decreasing on the interval that includes 2. Therefore, our answer is:

(1,9)

Report an Error

Example Question #3 : How To Find Decreasing Intervals By Graphing Functions

Is f(x) increasing or decreasing on the interval [4,7] ?

f(x)=-5x^4+6x^2-5x+3

Possible Answers:

The function is neither increasing nor decreasing on the interval [4,7] .

Decreasing, because the first derivative of f'(x) is negative on the function [4,7] .

Decreasing, because the first derivative is positive on the interval [4,7] .

Increasing, because the first derivative is positive on the interval [4,7] .

Increasing because the second derivative is positive on the interval [4,7] .

Correct answer:

Decreasing, because the first derivative of f'(x) is negative on the function [4,7] .

Explanation:

To find the an increasing or decreasing interval, we need to find out if the first derivative is positive or negative on the given interval. So, find f'(x) by decreasing each exponent by one and multiplying by the original number.

f(x)=-5x^4+6x^2-5x+3

f'(x)=-20x^3+12x-5

Next, we can find f'(4) and f'(7) and see if they are positive or negative.

$f'(4)=-20\cdot4^3+12\cdot4-5=-1237$

$f'(7)=-20\cdot7^3+12\cdot7-5=-6781$

Both are negative, so the slope of the line tangent to f(x) is negative, so f(x) is decreasing.

Report an Error

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Is f(x) increasing or decreasing on the given interval? How do you know?

f(x)=4x^3-12x^4+6x^5-4x^2+6x^3

[1,3]

Possible Answers:

Increasing, because f'(x) is positive on the interval [1,3] .

There is not enough information to tell whether f(x) is increasing or decreasing on the interval [1,3] .

Decreasing, because f'(x) is negative on the interval [1,3] .

Decreasing, because f''(x) is negative on the interval [1,3] .

Increasing, because f''(x) is positive on the interval [1,3] .

Correct answer:

Increasing, because f'(x) is positive on the interval [1,3] .

Explanation:

Recall that a function is increasing at a point if its first derivative is positive, and a function is decreasing if its first derivative is negative at that point. Therfore, we should start by finding f'(x). However, I will start by combining like terms and putting f(x) in standard form:

f(x)=4x^3-12x^4+6x^5-4x^2+6x^3

f(x)=6x^5-12x^4+10x^3-4x^2

f'(x)=30x^4-48x^3+30x^2-8x

Next, plug in each of our endpoints to see what the sign of f'(x) is.

f'(1)=30-48+30-8=4

$f'(3)=30\cdot 3^4-48\cdot 3^3+30\cdot 3^2-8\cdot 3=1380$

So f'(x) is positive on the given interval, so we know that f(x) is increasing on the given interval.

Report an Error

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Find the intervals where the following function is decreasing.

$y=\frac{2}{3}x^3 + x^2 -24x +2$

Possible Answers:

$(-4,\infty)$

$(3,\infty)$

$(-\infty,\infty)$

$(-\infty,-4)$

(-4,3)

Correct answer:

(-4,3)

Explanation:

The first step is to find the first derivative.

y'=2x^2 +2x -24

We can factor a out to get

y'=2(x^2+x-12) .

Now we need to solve for when y'=0 to get the critical points. Notice how factoring the 2 made the expression a little easier to simplify.

y'=2(x-3)(x+4)

x=3,-4.

The final step is to try points in all the regions $(-\infty,-4) , (-4,3), (3,\infty)$ to see which range gives a negative value for .

If we plugin in a number from the first range into , i.e , we get a positive number.

From the second range, , we get a negative number.

From the third range, , we get a positive number.

So the second range gives us values where the function is decreasing because is negative during that range, so (-4,3) is the answer.

Report an Error

Example Question #1 : Intervals

Regions problem

On which interval is the function shown in the above graph strictly decreasing?

Possible Answers:

Interval E

Interval B

Interval C

Interval D

Interval A

Correct answer:

Interval E

Explanation:

A function f(x) is strictly decreasing on an inverval if, for any y>x in the interval, f(y)<f(x) (i.e the slope is always less than zero)

Interval E is the only interval on which the function shows this property.

Report an Error

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Let $f(x)=\frac{4}{3}x^3-9x$ .

On which open interval(s) is f(x) decreasing?

Possible Answers:

There are no open intervals on which the function is decreasing.

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

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

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

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

Correct answer:

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

Explanation:

f(x) is decreasing on intervals where f'(x)<0 .

First, differentiate f(x) .

f'(x)=4x^2-9

Then, find the values for x for which the derivative is negative by solving

0=4x^2-9

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

$x=\frac{3}{2}\textrm{ or} \frac{-3}{2}$ .

Next, test the intervals.

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

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

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

Test them by substituting values for x:

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

Substitute -2,

f'(-2)=4(-2)^2-9=16-9=7

f'(-2)=7>0 .

The function is increasing on this interval since the derivative is positive on this interval.

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

Substitute 0,

f'(0)=4(0)^2-9=0-9=-9

f'(0)=-9<0 .

The function is deecreasing on this interval since the derivative is negative on this interval.

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

Substitute 2,

f'(2)=4(2)^2-9=16-9=7

f'(2)=7>0 .

The function is increasing on this interval since the derivative is positive on this interval.

Thus, $\left(-\frac{3}{2}, \frac{3}{2}\right)$ is the only interval on which the function is decreasing.

Report an Error

Example Question #3 : Decreasing Intervals

On what interval(s) is the function $g(t)=\frac{t}{(t^2+1)^2}$ decreasing?

Possible Answers:

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

$\left( \frac{1}{\sqrt{3}},\infty\right)$

$\left(-\infty, -\frac{1}{\sqrt{3}}\right), \left(\frac{1}{\sqrt{3}}, \infty\right)$

(-1,1)

$(-\infty, -1), (1, \infty)$

Correct answer:

$\left(-\infty, -\frac{1}{\sqrt{3}}\right), \left(\frac{1}{\sqrt{3}}, \infty\right)$

Explanation:

The function is decreasing when the first derivative is negative. We first find when the derivative is zero. To find the derivative, we apply the quotient rule,

$f'(x)=\frac{(t^2+1)1-t2(t^2+1)2t}{(t^2+1)^4}=\frac{(t^2+1)(1-3t^2)}{(t^2+1)^4}$ .

Therefore the derivative is zero at $t=\pm \frac{1}{\sqrt{3}}$ . To find when it is negative plug in test points on each of the three intervals created by these zeros.

For instance,

$f'(-1)=-\frac{1}{4}, f'(0)=1, f'(1)=-\frac{1}{4}$ .

Hence the function is decreasing on

$\left(-\infty, -\frac{1}{\sqrt{3}}\right), \left(\frac{1}{\sqrt{3}}, \infty\right)$ .

Report an Error

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

For which values of is the function $f(x)=x^4 -\frac{4}{3}x^3 -12x^2 +3$ decreasing?

Possible Answers:

-3 < x < 0 and $2 < x < \infty$

$-\infty < x < -2$ and 0 < x < 3

-2 < x < 0 and $3 < x < \infty$

This function is never decreasing.

$-\infty < x < -3$ and 0 < x < 2

Correct answer:

$-\infty < x < -2$ and 0 < x < 3

Explanation:

To determine where the function is decreasing, differentiate it:

f'(x)=4x^3 - 4x^2 -24x

What we are interested in are the points where f'(x)=0 . To determine these points, factor the equation:

0=4x(x-3)(x+2) this has solutions at x=-2, 0, 3

This splits the graph into 4 regions, and we can test points in each to determine if f'(x) is greater than or less than 0. If it is less than zero, the function is decreasing.

$- \infty < x < -2: f'(-3)=-72$ negative/decreasing

-2 < x < 0: f'(-1)= 16 positive/increasing

0 < x < 3: f'(1)=-24 negative/decreasing

$3 < x < \infty: f'(4)=96$ positive/increasing

Report an Error

Example Question #3 : Intervals

For which values of is the function $g(x)=\frac{1}{5}x^5-3x^3$ decreasing?

Possible Answers:

$- \infty < x < -3$ and $3 < x < \infty$

-3 < x < 3 , $x \neq 0$

$- \infty < x < -3$ and 0 < x < 3

This function is never decreasing.

-3 < x < 0 and $3 < x < \infty$

Correct answer:

-3 < x < 3 , $x \neq 0$

Explanation:

The function is decreasing where g'(x)<0 . To determine where this is happening, differentiate the function and find where g'(x)=0 . This will split the function into intervals where it is either increasing or decreasing.

g'(x)=x^4 - 9x^2

To determine where this equals zero, factor:

0 = x^2(x^2-9)=x^2(x-3)(x+3) this has solutions for x=-3, 0, 3 .

Test a point in each region to determine if it is increasing or decreasing within these bounds:

$- \infty < x < -3: g'(-4)=112$ positive/increasing

-3 < x < 0: g'(-1)=-8 negative/decreasing

0 < x < 3: g'(1)=-8 negative/decreasing

$3 < x < \infty: g'(4)=112$ positive/increasing

Report an Error

View Calculus Tutors

Jonathan
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Mathematics.

View Calculus Tutors

Steven
Certified Tutor

University of Louisville, Bachelor of Science, Mechanical Engineering. University of Phoenix-Louisville Campus, Master of Eng...

View Calculus Tutors

Jose
Certified Tutor

State University of New York at Albany, Bachelor of Science, Applied Mathematics. Teachers College at Columbia University, Ma...

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Chemistry Tutors in Boston, Physics Tutors in Washington DC, French Tutors in Boston, English Tutors in Philadelphia, Algebra Tutors in Houston, LSAT Tutors in San Diego, Algebra Tutors in Chicago, Math Tutors in Denver, English Tutors in Atlanta, Chemistry Tutors in Los Angeles

Popular Courses & Classes

SAT Courses & Classes in New York City, GRE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Miami, ACT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Houston, MCAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Boston

Popular Test Prep

GMAT Test Prep in Chicago, GMAT Test Prep in Denver, GRE Test Prep in Dallas Fort Worth, MCAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Denver, LSAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Atlanta, LSAT Test Prep in Los Angeles, MCAT Test Prep in Seattle, SSAT Test Prep in Washington DC

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

Calculus 1 : Decreasing Intervals

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : Intervals

Example Question #2 : Intervals

Example Question #3 : How To Find Decreasing Intervals By Graphing Functions

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Example Question #1 : Intervals

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Example Question #3 : Decreasing Intervals

Example Question #2 : How To Find Decreasing Intervals By Graphing Functions

Example Question #3 : Intervals

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: