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Question

Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.
$f(x)=\frac{x}{x^{2}+2}$

Carson Merrill · Accepted Answer

So here was he at the graph the interval of increase or would be from a point native, 1.414 to 1.41 For if we look at the graph of the inner the derivative, it&#x27;s clear here that it&#x27;s that is well, I&#x27;m And then the intervals of, uh, decrease would be negative. Infinity to maybe infinity to negative 1.4 and four and then 1.414 to positive infinity for concave ity. We see that the, um that from about from native 1.414 since this whole time, Um, it appears that the slope is positive. Um, it seems con cave up here up until we reached a zero point. So from negative infinity to one or 20 it&#x27;s Kong cave up. And then from zero to positive infinity, its concave down

Carson Merrill · Answer

so looking at her graph here, it appears that, um f is increasing on the interval from Ah, 0.382 Um, if you look here, it&#x27;s increasing here. So 0.32 toe infinity, it appears, um, it might be helpful to look at this graph here to determine the case. Eso you as we see at 0.3 to its increasing the graph is increasing. And then from negative infinity 2.3 it to the graph is decreasing. In terms of the cavity, we see that it&#x27;s conclave up. Um, it&#x27;s concave, up from zero 2.775 And it&#x27;s con cave down from negative infinity to zero and 0.775 to infinity. So the inflection points would be zero and 00.775

Carson Merrill · Answer

so looking at this graph here, we see, um, that it&#x27;s ah, increasing from infinity to zero. And then it&#x27;s decreasing from zero to negative or affirmative affinities. Here it&#x27;s increasing from infant zero to infinity, it&#x27;s decreasing. It is, uh, Kong Cave from negative Infinity toe one. And then it is Kong cave down from negative one to positive one. And then, from then on it&#x27;s Khan gave up from one to infinity, which tells us that the inflection points are negative one and positive one.

Carson Merrill · Answer

this graph here as we see as increasing from negative infinity Positive Indy. So there is no interval of decrease. We also see that, um, there is Khan cavity at Kong Cave, up from zero to Infinity Kong cave down from negative infinity to zero, so zero would be the only inflection point.

Carson Merrill · Answer

So if we look at this graph, we see that it&#x27;s increasing from negative 1 to 0 and then from positive one to infinity. And it&#x27;s decreasing from negative infinity to negative one and then zero toe one in terms of being, uh, connectivity, it&#x27;s Kong cave up from conclave up from negative infinity. If we look at this graph here we see it xcom cave, uh, from negative infinity to positive infamy. And as clear here is, just there&#x27;s this. It&#x27;s not differential at this point right here causes the sharp turn. Um, so I tells us that there is no inflection point and there&#x27;s no areas where it&#x27;s conch eight down.

Carson Merrill · Answer

So who received a graph of our function and we see that it&#x27;s increasing. If we look at the, uh, derivative function might be easier to tell. We see it&#x27;s increasing from negative 0.5 to infinity, and it&#x27;s decreasing from negative infinity to negative 0.5. Um, the graph is conclave up from negative to to positive one. Um, and the graph is con cape down from negative infinity too negative too, and from one to infinity. So it tells us that our inflection points would be negative to positive one.

Problem

Find: (a) the intervals on which f is increasing,…

Problem 22 Easy Difficulty

Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.
$f(x)=\frac{x}{x^{2}+2}$

Answer

A. $f$ is increasing when $f^{\prime}$ is positive
B. f is decreasing when $f^{\prime}$ is negative
C. $f$ is concave up when $f^{\prime \prime}$ is positive
D. $f$ is concave down when $f^{\prime \prime}$ is negative
E. Therefor inflection occurs at $x=0, \pm \sqrt{6}$

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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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A. $f$ is increasing when $f^{\prime}$ is positiveB. f is decreasing when $f^{\prime}$ is negativeC. $f$ is concave up when $f^{\prime \prime}$ is positiveD. $f$ is concave down when $f^{\prime \prime}$ is negativeE. Therefor inflection occurs at $x=0, \pm \sqrt{6}$

Calculus Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Catherine R.

Anna Marie V.

Kristen K.

A. $f$ is increasing when $f^{\prime}$ is positive
B. f is decreasing when $f^{\prime}$ is negative
C. $f$ is concave up when $f^{\prime \prime}$ is positive
D. $f$ is concave down when $f^{\prime \prime}$ is negative
E. Therefor inflection occurs at $x=0, \pm \sqrt{6}$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals