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.$$f(x)=3 x^{5}+5 x^{4}-20 x^{3}+10 x+30$$

$\mathrm{CU}$ if $-2<x<0$ or $x>1 ; \mathrm{CD}$ if $x<-2$ or $0<x<1 \mathrm{I}(-2,154),(0,30)$ and (1,28)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Oregon State University

University of Nottingham

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.

04:13

Determine where the functi…

03:48

01:16

01:50

01:08

Determine the intervals on…

01:54

00:41

06:48

Concavity Determine the in…

02:14

01:11

01:29

Find the intervals on whic…

0:00

Find the intervals on whi…

question 28 would like you to determine where the function F of X equals three x to the fifth plus five x to the fourth minus 20 x to the third plus 10. X plus 30 is concave up or down and list inflection points. Uh, so to do so, you need to find the second derivative of F of X. So F prime of X would be equal to 15 x to the fourth plus 20 x to the third, minus 60 X squared plus 10 f double prime of X, then would be 60 x to the third plus 60 x squared minus 1 20 x This then, to find our inflection points we need to find when f the prime of X is equal to zero. Ah, we can first start by setting that equal to our equation, but simplifying it so 60 x times X squared plus X minus two that is unequal to 60 x times X plus two times X minus one. Therefore, alphabet F double prime of X is equal to zero when X is equal to negative 20 and one. Ah, then finding those inflection points by plugging in these values into F of X. You would get negative to 154 0 30 and one and 28. So those are all your inflection points now. You would need to test intervals ah, to find whether it's concave up or down. So when X is less than negative two, you can test f double prime of negative three, which is negative. 7. 20. So you're a con cave down when X is less than negative. Two. Ah testing when X is between negative too, and zero after the prime of negative one is equal to 1 20. So Khan gave up between negative two and zero when X is in between zero and one F double prime of one half is equal to negative 37.5. So calm gave down between zero and one. And lastly, when X is greater than one F, double prime of two is 4 80. Therefore, your concave up when X is greater than one, and these are all your answers for Question 28

View More Answers From This Book

Find Another Textbook

Prove the quotient rule using the same kind of trick that was used for the p…

07:25

Minimize the amount of fencing needed to construct the region shown in Figur…

01:06

Use the first and second derivatives to sketch the graph of the given equati…

01:12

Show that $a x^{2}+b x+c$ always has the same sign as the constant term $c$ …

04:46

(a) Find $d y$ if $y=\frac{3 x-1}{2 x-3}$,(b) Find $f^{\prime}(x)$ if $f…

01:04

Determine the number of real roots of the equation $x^{3}+3 x^{2}-9 x-15=0$<…

03:13

Differentiate $\left(2 x+x^{4}\right)^{2}$ with respect to $x^{3} .$ Hint: l…

01:13

(a) Given $f(x)=x^{2}-4,$ and $g(x)=\left(x^{2}-4\right)^{10} .$ Find the po…

03:38

Determine the extrema on the given interval.$$f(x)=x /\left(x^{2}+1\righ…

01:15

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