4.5 Derivatives and the shape of a graph (Page 6/14)

Calculus volume 1 Page 6 / 14

Consider the function $f (x) = x^{3} - (\frac{3}{2}) x^{2} - 18 x .$ The points $c = 3, −2$ satisfy $f^{'} (c) = 0 .$ Use the second derivative test to determine whether $f$ has a local maximum or local minimum at those points.

$f$ has a local maximum at $−2$ and a local minimum at $3 .$

Got questions? Get instant answers now!

We have now developed the tools we need to determine where a function is increasing and decreasing, as well as acquired an understanding of the basic shape of the graph. In the next section we discuss what happens to a function as $x \to ± \infty .$ At that point, we have enough tools to provide accurate graphs of a large variety of functions.

Key concepts

If $c$ is a critical point of $f$ and $f^{'} (x) > 0$ for $x < c$ and $f^{'} (x) < 0$ for $x > c,$ then $f$ has a local maximum at $c .$
If $c$ is a critical point of $f$ and $f^{'} (x) < 0$ for $x < c$ and $f^{'} (x) > 0$ for $x > c,$ then $f$ has a local minimum at $c .$
If $f ″ (x) > 0$ over an interval $I,$ then $f$ is concave up over $I .$
If $f ″ (x) < 0$ over an interval $I,$ then $f$ is concave down over $I .$
If $f^{'} (c) = 0$ and $f ″ (c) > 0,$ then $f$ has a local minimum at $c .$
If $f^{'} (c) = 0$ and $f ″ (c) < 0,$ then $f$ has a local maximum at $c .$
If $f^{'} (c) = 0$ and $f ″ (c) = 0,$ then evaluate $f^{'} (x)$ at a test point $x$ to the left of $c$ and a test point $x$ to the right of $c,$ to determine whether $f$ has a local extremum at $c .$

If $c$ is a critical point of $f (x),$ when is there no local maximum or minimum at $c ?$ Explain.

Got questions? Get instant answers now!

For the function $y = x^{3},$ is $x = 0$ both an inflection point and a local maximum/minimum?

It is not a local maximum/minimum because $f^{'}$ does not change sign

Got questions? Get instant answers now!

For the function $y = x^{3},$ is $x = 0$ an inflection point?

Got questions? Get instant answers now!

Is it possible for a point $c$ to be both an inflection point and a local extrema of a twice differentiable function?

Got questions? Get instant answers now!

Why do you need continuity for the first derivative test? Come up with an example.

Got questions? Get instant answers now!

Explain whether a concave-down function has to cross $y = 0$ for some value of $x .$

False; for example, $y = \sqrt{x} .$

Got questions? Get instant answers now!

Explain whether a polynomial of degree $2$ can have an inflection point.

Got questions? Get instant answers now!

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where $f$ is increasing or decreasing.

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−2, 0). Then it continues increasing a little before decreasing and crossing the x axis at (−1, 0). It achieves a local minimum at (1, −6) before increasing and crossing the x axis at (2, 0).

Increasing for $−2 < x < −1$ and $x > 2;$ decreasing for $x < −2$ and $−1 < x < 2$

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts negative and touches the x axis at the origin. Then it decreases a little before increasing to cross the x axis at (1, 0) and continuing to increase.

Decreasing for $x < 1,$ increasing for $x > 1$

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts positive and decreases to touch the x axis at (−1, 0). Then it increases to (0, 4.5) before decreasing to touch the x axis at (1, 0). Then the function increases.

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts at (−2, 0), decreases to (−1.5, −1.5), increases to (−1, 0), and continues increasing before decreasing to the origin. Then the other side is symmetric: that is, the function increases and then decreases to pass through (1, 0). It continues decreasing to (1.5, −1.5), and then increase to (2, 0).

Decreasing for $−2 < x < −1$ and $1 < x < 2;$ increasing for $−1 < x < 1$ and $x < −2$ and $x > 2$

Got questions? Get instant answers now!

For the following exercises, analyze the graphs of $f^{'},$ then list all intervals where

$f$ is increasing and decreasing and
the minima and maxima are located.

The function f’(x) is graphed. The function starts at (−2, 0), decreases for a little and then increases to (−1, 0), continues increasing before decreasing to the origin, at which point it increases.

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts at (−2, 0), increases and then decreases to (−1, 0), decreases and then increases to an inflection point at the origin. Then the function increases and decreases to cross (1, 0). It continues decreasing and then increases to (2, 0).

a. Increasing over $−2 < x < −1, 0 < x < 1, x > 2,$ decreasing over $x < −2,$ $−1 < x < 0, 1 < x < 2;$ b. maxima at $x = −1$ and $x = 1,$ minima at $x = −2$ and $x = 0$ and $x = 2$

Got questions? Get instant answers now!

The function f’(x) is graphed from x = −2 to x = 2. It starts near zero at x = −2, but then increases rapidly and remains positive for the entire length of the graph.

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts negative and crosses the x axis at the origin, which is an inflection point. Then it continues increasing.

a. Increasing over $x > 0,$ decreasing over $x < 0;$ b. Minimum at $x = 0$

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing a little before decreasing and touching the x axis at the origin. It increases again and then decreases to (1, 0). Then it increases.

Got questions? Get instant answers now!

For the following exercises, analyze the graphs of $f^{'},$ then list all inflection points and intervals $f$ that are concave up and concave down.

The function f’(x) is graphed. The function is linear and starts negative. It crosses the x axis at the origin.

Concave up on all $x,$ no inflection points

Got questions? Get instant answers now!

The function f’(x) is graphed. It is an upward-facing parabola with 0 as its local minimum.

Got questions? Get instant answers now!

The function f’(x) is graphed. The function resembles the graph of x3: that is, it starts negative and crosses the x axis at the origin. Then it continues increasing.

Concave up on all $x,$ no inflection points

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−0.5, 0). Then it continues increasing to (0, 1.5) before decreasing and touching the x axis at (1, 0). It then increases.

Got questions? Get instant answers now!

The function f’(x) is graphed. The function starts negative and crosses the x axis at (−1, 0). Then it continues increasing to a local maximum at (0, 1), at which point it decreases and touches the x axis at (1, 0). It then increases.

Concave up for $x < 0$ and $x > 1,$ concave down for $0 < x < 1,$ inflection points at $x = 0$ and $x = 1$

Got questions? Get instant answers now!

For the following exercises, draw a graph that satisfies the given specifications for the domain $x = [−3, 3] .$ The function does not have to be continuous or differentiable.

$f (x) > 0, f^{'} (x) > 0$ over $x > 1, −3 < x < 0, f^{'} (x) = 0$ over $0 < x < 1$

Got questions? Get instant answers now!

$f^{'} (x) > 0$ over $x > 2, −3 < x < −1, f^{'} (x) < 0$ over $−1 < x < 2, f ″ (x) < 0$ for all $x$

Answers will vary

Got questions? Get instant answers now!

$f ″ (x) < 0$ over $−1 < x < 1, f ″ (x) > 0, −3 < x < −1, 1 < x < 3,$ local maximum at $x = 0,$ local minima at $x = ± 2$

Got questions? Get instant answers now!

There is a local maximum at $x = 2,$ local minimum at $x = 1,$ and the graph is neither concave up nor concave down.

Answers will vary

Got questions? Get instant answers now!

Questions & Answers

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

treat them normal, if they want help then give them. that will make everyone happy

Saurabh

What are the treatment for autism?

Magret Reply

hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these

Jharna

continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child

Jharna

child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective

Jharna

results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.

Jharna

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 7

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask

©flickr:	Microbiology Final By Szilárd Jankó Start Quiz
©flickr: Urko	Music Reviewer MCQ By Edgar Delgado Start Quiz
	Anthropology Language Culture By Richley Crapo Start Assignment
	8 Biology 08 Photosynthesis MCQ By OpenStax Start Quiz
	Social Dances 1 By Marion Cabalfin Start Test
	Concepts of biology By OpenStax Read Online Course
	3 Neuroscience Exam 2004 4 By David Corey Start Exam
	Social Work midterm By Katy Pratt Start Exam
	26 Toxicology Gustafson- Biotoxins and Venoms By Brooke Delaney Start Exam
©flickr: BK	Self Confidence By Miranda Reising Start Quiz