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(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the inforvation from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.$f(x)=2+2 x^{2}-x^{4}$

A. increasing at $[-\infty,-1] U[0,1] ;$ decreasing at $[-1,0] U[1, \infty]$B. local min. $(0,2), \quad$ local max: $(-1,3),(1,3)$C. conc.down: $\left(-\infty,-\frac{1}{\sqrt{3}}\right),\left(\frac{1}{\sqrt{3}}, \infty\right), \quad$ conc.up: $\left(-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right), \quad$ inf.pts $\left( \pm \frac{1}{\sqrt{3}}, \frac{23}{9}\right)$D. SEE GRAPH

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Missouri State University

Campbell 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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) Find the intervals of …

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(a) Find the intervals…

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for this question. We have four different parts of in the beginning. We just ride it out of the first so that they were there and then a stakeholder narrative. And so we write out this from solidarity. If I can just do the anemia Factoring, yes, this. So if we write out in this way, it's easier to figure out the roots of this if prime. So if the prime that's taken out of your day vehicles do minus 12 exits were class four. Okay, so part eight, we're going to figure out the increasing and decreasing interval. So you just said if prime of X equals zero so we have three different Rotech Sequels to minus 10 in the one on these three points splits the whole interval, the whole dorm mate with you till four different parts. So the first interface from my a negative infinity tone active one. So over this imperil prime affects, it's uniformly positive. We can choose at every point inside this interval in the Prague in tow s prime, and they we can very five Mrs always positive. So it's increasing the function. Is the increasing over this in trouble? Next intervals from minus 1 to 0. So I only see terrible. The force of the narrative is negative since Deke Racing, and we do the same thing for the next two from zero to wander from one to positive infinity so we can check off this result by parking some specific value united in trouble. So we have to increasing in Tebow and the toothy crazy interval. Not for part B, part B. We want to find out a local maximum local minimum which is related to so for Patty we know. So this locates the first point. X equals to one X equals minus one. So Onal on the left, it's increasing on all the rightist secrets. Meh means we have a local maximum het X equals to one SRE minus one because the increasing for us in decreasing So the value we will be three and we also has also have another local makes that exceed coast. One in the value is also three for the same reason. It's increasing for us than the equations. There's a peak in a full of local immune. We have one. No commitment attacks equals zero. The value will be too because it's decreasing for US 10 million crazy and the for the Integrity Passy connectivity product. We need to use the second intuitive and you just said the second narrative because it's euros. So we have X equals two pass minus. Who told 1/3 so have to tweet different points and this pleats little main 23 prods So first product is from minus infinity to minus root off 1/3. So over this interval of the second of curative yes, negative. So it's conclave I don't and the for a second over from minus root off 1/3 to route all winter A steak on the curative is positive So it's conch A farm on the phone for the last one Wrong route off 1/3 toe Positive infinity The second of purity of face Conectiv six come case down again and the foot from here We can see we have to your collection points. Oh, it's so why is X minus root off 1/3 And, uh, quality of brutal print there because the continuities um how different at this point? So for off from this information, we can actually sketch the function which is party, so if you want to sketch your function, We need to lay both some specific point, like minus 10 and poor people one and Reza label this minus brutal. Want the other in the root of 1/3 here. So we can see for one and X equals toe one minus one. The value here. It should be three by our argument before. And, uh, the Y intercept is too. So the graph looks like this. So don't forget, we have to change it. A cantare ET That's miss two points. And we can see we have some critical points year year, and hear off them. Have horizontal tension. Lies. Um, so that's it.

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