Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Use the guidelines of this section to sketch the curve.$$y=x^{4}-4 x$$

$(-\infty, \infty)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

Missouri State University

Campbell University

Baylor University

Idaho State University

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

11:07

Use the guidelines of this…

17:51

17:18

17:53

12:42

14:58

05:43

16:02

13:27

07:53

we're given the equation. F event is equal to X four minus four. Then we have to sketch first. We can figure out the domain of this function and the domain of this functions and doesn't have an irrational or square root aspects of it. It is just negative. Infinity to infinity. There are no restrictions. You can also say that it is for all real numbers. What are the intercepts we could talk about? First is the X interest. So we can set this equation equal to zero, and we can factor out minus four. So we have an X intercept already. And then for this part, we have X cubed minus 400 And that is equal to the cube root of four. So the cube root four zero. So what about X intercept? Well, if I plug in zero for X, then we get zero minus zero. So the Y intercept is also 00 or the origin. We can perform the symmetry test so we can see if it is. Even if the original equation is the same thing as if we were placed acts with negative. We have negative X to the fourth power minus four times negative. We left with to the fourth less for this is not this original equation. It is not even. So. What about the Well, if we negate the entire function, Is that the same thing as what we got from the from the even? So what happens when we negate the entire function? We get negative X to the fore plus four X And these two functions are not the same thing because this is negative for negative four. This is positive. So it is not odd as well. So there is not even not on. And we could, um the next step is asking toes, but there are no ask him to its any polynomial function next step, since we were all this is from original function. To find more information, we have to find the first derivative, which is four X cubed minus four. And we could set this equal to zero to find any minimum or maximum values. So four X cubed minus four equal to zero we solve for X, we get remove This four over here, divided by four x cubed is equal to one or X is just equal to one And if we plug this back into the original equation so we have our X value are y value is one minus four. So that's negative. Three. Since we have this value of one, we can figure out being intervals we contest. So we have negative infinity to one and then we have one to infinity to find out if it is increasing or decreasing. So we have our we can have a test value. We can have our sign. So for this test volume, you can say that's a zero for this one. We can just say to So if we plug in zero to our the first derivative equation yet a value of zero minus four So that is negative to get too cute is a and that's obviously going to be positive. So it will be decreasing on this interval, decreasing and they'll be increased on this. So we said it was increasing on one to infinity and decreasing on negative infinity to one So and next thing we can check is concave ity. So we need to figure out the second derivative one of the river of this function which is just 12 x squared. We could find points of inflection. So we set this equal to zero. But then it is just going to be X is equal to zero. As a potential points of inflection, we could figure out intervals length negative infinity 20 to figure out the concave ity of these intervals. But notice that the second derivative is 12 x squared. This function will always be greater than or equal to zero for all X. No matter what value of extra potency here, it will be greater than equal to zero. So that means the concave ity the sign of whenever test value will be positive here, and it will be positive here. So if you chose negative one for this and if we chose that, say one for here, it would be positive, and this will be positive as well. So this is actually not a point of inflection. Oh, so you can just list here No point of inflection. The last thing we have to do is actually sketch the ground. So you know a few things about this craft? No, it's X centers up. I'll draw it in a blue. Oh, this one is at 00 this one is at the cube root of four zero. And then we have our what looks like a a minimum because it is decreasing from here from negative infinity toe. One, 21 and you'll be increasing. So try to draw. That would look like he thought that like this, that is our graph f of

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Use the guidelines of this section to sketch the curve.

$ y = x^4 - …

$ y = \frac{…

$ y = x(x - …

$ y = (4 - x…

$ y = (x - 4…

$ y = e^x/x^…

Use the guidelines of this section to sketch the curve.$$y=x e^{-x}$$

$ y = e^{2x}…

06:29

A roast turkey is taken from an oven when its temperaturehas reached $18…

01:09

Find the derivative. Simplify where possible.$f(t)=\operatorname{sech}^{…

01:07

Sketch the graph of a function that satisfies all of the given conditions.

04:17

For what values of $a$ and $b$ is the following equation true?$$\lim _{x…

04:45

Strontium-90 has a half-life of 28 days.(a) A sample has a mass of 50 $\…

01:06

Differentiate the function.

$$f(x)=\left(x^{3}+2 x\right) e^{x}<…

01:29

Find the most general antiderivative of the function(Check your answer b…

00:47

$1-38=$ Find the limit. Use l'Hospital's Rule where appropriate.

02:51

(a) Find the intervals on which $f$ is increasing or decreasing.(b) Find…

03:56

The graph of the derivative $f^{\prime}$ of a continuous function $f$ is sho…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.