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 and second derivatives to sketch the graph of the given equation. Also include the intercepts, whenever they are easily determined.$$f(x)=x^{4}-24 x^{2}+25$$

$$m \text { at }(\pm 2 \sqrt{3},-119)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

03:32

Use the first and second d…

03:07

02:48

01:44

02:02

01:23

01:11

03:41

02:57

03:05

02:30

01:06

09:56

01:12

03:33

03:14

02:53

01:30

03:01

01:21

question 52 would like you to use the 1st and 2nd derivative to start to sketch the graph of f of X equals X to the fourth minus 24 X squared plus 25 include any intercepts if they're easy to find. Uh, so starting off you can take F prime of X to find the critical points. So at promo bags is equal 24 x cubed minus 48 x sending not equal to zero would be forex times X squared minus 12 is equal to zero. Uh, so Exton is equal to zero and plus or minus two square three, which is about 3.5 for when we're graphing. Um, And then you can plug those points into F uh, so you can plot them so f of negative to screw. Three is negative. 1 19 f of zero is 25 f of negative to score positive to score three. It's also a negative 1 19, now taking F double prime of X that is just equal to 12 x squared, minus 48 plugging in our critical points to see if their maximum or minimum after the crime of negative two squared three is 96 so concave up is a minimum after the prime of zero is negative. 48. So crank it down. It's a maximum. And after the prime of two square three is 96. So also concave up and a minimum. Now you can use all that information to grab. So you have they 0.0 25 and you know you're going all the way down to negative 1 19 at about 3.5 and negative 3.5. So putting those points on our graph and from there knowing that those two, these two are minimums and that is a maximum, your graph would look something like this when you connect those points, and that's your answer to question 52.

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:46

How much should be deposited into an account today if it is to accumulate to…

01:05

Show that $a_{\mathrm{k}}=\frac{f^{(k)}(0)}{k !}$ where the symbol $$k !=k(k…

02:03

Explain the relationship between the answers in parts (a) and (b) of Exercis…

01:55

Sketch the graph of the function defined by the given equation.$$f(x)=3\…

02:37

Find the equation of the tangent line to the curve at the given $x$ -value.<…

04:45

(Continuation of previous exercise.) Suppose that the average price of a VCR…

09:54

In Exercises $29-38,$ for the function determined by the given equation (a) …

01:16

Determine where the function is concave upward and downward, and list all in…

01:14

If $f(x)=(1+x)^{5}=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+a_{5} x…

04:04

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