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Q. 33

Expert-verified

Calculus

Found in: Page 261

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$l n (x^{2} + 1)$

Ans:

After inserting the root values we can find the increasing and decreasing intervals of the given function.

Intervals of the given function are

Increasing at $(0, \infty)$

Decreasing at $(- \infty, 0)$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information:

$l n (x^{2} + 1)$

Step 2. Finding the derivative of the function:

$f (x) = \ln (x^{2} + 1) f^{'} (x) = \frac{2 x}{x^{2} + 1} l e t f^{'} (x) = 0 ∴ \frac{2 x}{x^{2} + 1} = 0 \Rightarrow 2 x = (x^{2} + 1) 0 \Rightarrow 2 x = 0 x = 0$

Step 3. Inserting the root points on the number line(Sign chart):

After inserting the root values we can find the increasing and decreasing intervals of the given function.

Intervals of the given function are

Increasing at $(0, \infty)$

Decreasing at $(- \infty, 0)$

Step 4. Verifying algebraic answers with graphs :

Most popular questions for Math Textbooks

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = \frac{x^{2} - x}{x^{2} - 3 x + 2}$

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

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