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

Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=4 x^{3}-6 x^{2}+2 x+1$$

$$\mathrm{CD} \text { if } x<1 / 2 ; \mathrm{CU} \text { if } x>1 / 2 \mathrm{I}(1 / 2,1)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Oregon State University

Harvey Mudd College

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.

02:05

Determine where the functi…

01:13

00:41

00:44

02:58

Determine where the graph …

01:11

01:29

02:00

01:49

01:03

Use the Concavity Theorem …

01:54

02:06

Find the intervals on whic…

01:02

01:16

Question 25 wants you to determine where the function F of X equals four X cubed minus six X squared plus two X plus one is concave up or down and list any inflection points. So to do so, we need to get to the second derivative. So F prime of X is 12 X squared minus 12 X plus two F double prime of X is equal to 24 x minus 12. So first we can find inflection points. I would be when f double prime of zero or of X is equal to zero. So setting your equation 24 x minus 12 equal to zero gives you X is equal to one half. Plugging that into FX F of one half is equal to once or inflection point is at one half one. Now, testing work on cavity is so testing for X. Less than one half take F double prime of a number like zero that gives you negative 12, which means that when X is less than one half, you're concave down and testing a number greater than one half so F double prime of one is equal to 12. Therefore, when access greater than one half, you are concave up, and those are your answers for question 25

View More Answers From This Book

Find Another Textbook

Numerade Educator

04:20

A 26 foot ladder is leaning against a vertical wall and its base is on level…

05:11

Suppose that the oil spill from the damaged hull of a ship forms a circular …

01:20

Use your calculator to compute the expression given.$$7^{-3.25}$$

01:53

Suppose that the manufacturer in Exercise 1 is forced to cut the selling pri…

01:15

Sketch the graph of the function defined in the given exercise. Use all the …

05:48

(a) Determine the $x$ -intercepts, (b) the vertical asymptotes, (c) the hori…

05:01

Find $d y / d x$ using any method.$$\frac{x^{2}-y^{2}}{x^{2}+y^{2}}=2 x+…

04:28

(a) $f(x)=\left(x^{2}+2\right)\left(x^{4}-7\right) .$ Find $d f.$(b) $y=…

02:08

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

02:11

Let $f(x)$ be defined on the closed interval [0,1] by the rule:$$f(x)=\l…