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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.$f(x)=\frac{x}{x^{2}+2}$

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 1 / AB

Chapter 4

THE DERIVATIVE IN GRAPHING AND APPLICATIONS

Section 1

Analysis of Functions I: Increase, Decrease, and Concavity

Functions

Limits

Derivatives

Differentiation

Continuous Functions

Applications of the Derivative

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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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's clear here that it's that is well, I'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's Kong cave up. And then from zero to positive infinity, its concave down

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