Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Use the first derivative to determine where the given function is increasing and decreasing.$g(x)=4 x^{3}-24 x^{2}+36 x+96$

Dec on $1 < x < 3 ;$ inc on $x < 1$ or $x > 3$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Harvey Mudd College

Baylor University

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

03:36

Use the first derivative t…

03:51

01:15

Use the Monotonicity Theor…

02:59

04:13

00:56

Use the derivative to help…

01:08

Determine where each funct…

03:04

This is Chapter three, Section two problems. Seven. And what this problem asks us to do is find out where our function G FX is increasing and where it is decreasing by finding the first derivative. So the first step is finding the critical points. And if you remember, the critical points are where the derivative of the function is equal to zero. So I already went ahead and took the derivative of G of X, which was g avec g prime of X. And from there I got 12 x squared minus 48 x plus 36. And to make it a little easier, I factored out the 12. And that gave me 12 times X squared minus four x plus three. Furthermore, that X squared minus four X plus three could be factored down again to get 12 times X minus one times X minus three. So that gave us critical points at X equals one and X equals three. From here, we can draw our number line so we have our critical point at X equals one, and we have our critical point at X equals three, and we're going to pick three numbers. We're going to pick a number that is less than X minus one. We're going to pick a number that is in between 13 and we're going to pick a number that is greater than three. And we're going to plug each of those numbers into G prime of X to see whether I function is increasing or decreasing in that area. Okay, so first, let's start with G Prime of Let's say that are Let's get some values. Okay, so let's say that our number less than one we can say X equals zero here. X equals two here in X equals for here. So we know that X equals one that equals zero and X equals three. That equals zero. So let's plug in other ones. So if we have X equals zero, we have g prime of zero. Yeah, that's going to look like 12 times zero minus one time, zero minus three. That'll be 12 times negative. One times negative three. And that is going to give us a positive 36 because it's positive. We know that our function is going to be increasing if X is less than one Mhm. So now, for our ex of two, which is in between one and three. Mhm. We'll have 12 times two minus one two minus three. Yeah. Yeah. So here we have 12 times, one times negative one. And that's going to give us a negative 12. Because it's negative. We know that the interval in between X equals one and X equals three. That is going to be decreasing. Yeah, Lastly, we're going to have G. Yeah, we're gonna have G prime of four. Our last value of X. That's gonna give us 12 times four minus one times four minus three. And that's gonna give us 12 times, three times one. And that's going to give us again a positive 36. Which means if we have an X value greater than three, our function is again going to be increasing. So to sum it all up, our function G FX it was going to be increasing when X is less than one or X is okay, greater than three. And it is going to be decreasing when X is between one and three. So here's your answer right here

View More Answers From This Book

Find Another Textbook

02:55

Find the point on the curve $y=x^{3}$ at which the tangent line at (2,8) cro…

04:04

If $y=x^{3}-\sqrt{2 x},$ find $d x / d t$ when $x=8$ and $d y / d t=64$.

02:10

Determine the coordinates of the midpoint of the line segment joining the po…

05:24

Determine the equation of the tangent line to the given curve at the indicat…

03:00

A right triangle is formed in the first quadrant by a line passing through t…

01:32

Compute the indicated limit.$$\text { (a) } \lim _{x \rightarrow 5^{+}} …

Find the point on the curve $y=\sqrt{x}$ closest to $(3 / 2,0)$.

03:23

Use the first derivative to determine where the given function is increasing…

02:33

Suppose that the manufacturer in Exercise 1 can manage to reduce the overhea…

01:47

Use the appropriate rules to determine the derivative.$$w=32 v^{1 / 4}-\…