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

Classify all critical points.$h(x)=4 x^{3}-13 x^{2}+12 x+9$

$$\mathrm{m}(3 / 2,45 / 4), \mathrm{M}(2 / 3,335 / 27)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Oregon State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

01:37

Locate all critical points…

03:04

Classify all critical poin…

03:53

04:05

02:31

01:55

02:55

02:16

Determine all critical poi…

01:12

02:09

02:33

01:59

01:50

wanna derive this equation. So we use this, we do this using power rule. So are the derivative of our first term is going to be 12 that squared -26 x plus 12. Now here we can actually factor out to at the very least and get six X squared one is 13 eggs plus six. Now using our grouping for factory mary Wake up, we can factor this. So we have 2X. That can come out of the front because We want something that multiplies to 36 and adds up to negative 13. And that gives us four and nine Made a foreign -9. So we can have it be two X. Or well I'll go one step backwards. So six X squared minus or x minus nine X Plus six. And then we can Factor things out like two x. here It was it's -2, three x -2 and three. Here to get a three X -2. So to solve each of these, we have our In total we have two x minus three and three x -2 As our functions for zero. And so we get X equals three halves and two thirds. Now we want to plug both of these in to our original function. Now let you plug and chug that on your own to get the watt corresponding why values so our corresponding why values? We'll be 45 or four and 3 35. Uh huh. 27. Again you wanna you do get that just by plugging in your X value to the original equation because that will actually give you the points now time to figure out if these are minimum or maximum. So to do this we have a number line and we can pick we want to pick values that are slightly above and slightly below each of these zeros. So two thirds 3/2, you can pick one here And let's say two and 0. So We know at each of these points the derivative equal zero. So looking at the derivative function again, right here at zero. Will this equation be positive or negative? And it will be negative and then at one, will it be positive or negative? Again? You don't need to have actual values. You just want to tell if the drift of itself is positive. Native to see if it's increasing or decreasing. So at one it's positive And two it's positive. So based on this information, based on this information, you can actually kind of graphic And that you have a positive derivative at zero positive derivative at one. Mm. Okay. Mhm. And then a negative derivative. So actually it's going to be negative. That's euro. And so we have a minimum at at three hats and a maximum at two thirds

View More Answers From This Book

Find Another Textbook

01:20

$$\text { Find } \frac{d^{2} y}{d x^{2}}, \text { if } x^{1 / 2}-y^{1 / 2}=6…

01:01

Draw the graph of a function such that the maximum is also a relative maximu…

03:17

Approximate, using the method of the previous exercise, $f(31.99)$, if $f(x)…

02:07

(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find t…

06:00

Use linearization to approximate the given quantity. In each case determine …

01:43

Locate all critical points.$$h(x)=4 x^{3}-13 x^{2}+12 x+9$$

06:45

The Amalgamated Flashlight Company shows a profit of $4,500$ on a production…

02:18

Find the average rate of change of $y$ with respect to $x$ on the given inte…

02:00

The shadow of a tree is 40 feet long at the same time that the shadow of a 4…

04:06

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