Q. 30 Use a sign chart for f'聽to dete... [FREE SOLUTION]

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Chapter 3: Q. 30 (page 261)

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.
$f (x) = \frac{3 x + 1}{x^{2} - 1}$

Short Answer

Expert verified

Ans:

Intervals of the given function are $(- 鈭�, \frac{- 1}{3}] 鈭� [\frac{- 1}{3}, 鈭�)$

Always Decreasing function.

Step by step solution

Step 1. Given information:

$f (x) = \frac{3 x + 1}{x^{2} - 1}$

Step 2. Finding the derivative of the function:

$f (x) = \frac{3 x + 1}{x^{2} - 1} f^{'} (x) = \frac{(x^{2} - 1) (3) - (3 x + 1) (2 x)}{{(x^{2} - 1)}^{2}} \frac{鈭�}{鈭� x} \frac{u}{v} = \frac{v \frac{鈭�}{鈭� x} (u) - u \frac{鈭�}{鈭� x} (v)}{v^{2}} f^{'} (x) = \frac{3 x^{2} - 3 - 6 x^{2} - 2 x}{{(x^{2} - 1)}^{2}} = \frac{- 3 x^{2} - 2 x - 3}{{(x^{2} - 1)}^{2}} l e t f^{'} (x) = 0 鈭� \frac{- 3 x^{2} - 2 x - 3}{{(x^{2} - 1)}^{2}} = 0 鈬� - 3 x^{2} - 2 x - 3 = {(x^{2} - 1)}^{2} 鈬� - 3 x^{2} - 2 x - 3 = x^{4} - 2 x^{2} + 1 鈬� x^{4} + x^{2} + 2 x + 2 x = \frac{- 1}{3}$

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 $(- 鈭�, \frac{- 1}{3}] 鈭� [\frac{- 1}{3}, 鈭�)$

Always Decreasing function.

Step 4. Verifying algebraic answers with graphs :

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91影视

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.
$f (x) = \frac{3 x + 1}{x^{2} - 1}$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Finding the derivative of the function:

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

Step 4. Verifying algebraic answers with graphs :

One App. One Place for Learning.

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91影视

Use a sign chart for f'to determine the intervals on which each function fis increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.f(x)=3x+1x2-1

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Finding the derivative of the function:

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

Step 4. Verifying algebraic answers with graphs :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

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.
$f (x) = \frac{3 x + 1}{x^{2} - 1}$