Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use the definitions of increasing and decreasing to argue that $f (x) = x^{4}$ is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ . Then use derivatives to argue the same thing.

The statement has been proven.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

We have been given a function $f (x) = x^{4}$ .

We have to use the definitions of increasing and decreasing to argue that this function is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ .

Step 2. Using the definition

For a and b in the interval $(- \infty, 0]$

Now if $a < b \leq 0$

Then,

role="math" localid="1648442267515" $a^{4} > b^{4}$

Thus the function is decreasing in the interval $(- \infty, 0]$

Also,

For a and b in the interval role="math" localid="1648442301380" $[0, \infty)$

Now if $0 < a < b$ then,

$a^{4} < b^{4}$

Thus the function is increasing in the interval $[0, \infty)$

Step 3. Using the derivative

The derivative of the function is given by :

$f^{'} (x) = 4 x^{3}$

The function $f^{'} (x) = 4 x^{3}$ is always negative for x<0

The function $f^{'} (x) = 4 x^{3}$ is always positive for x>0

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Calculus

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Short Answer

Use the definitions of increasing and decreasing to argue that $f (x) = x^{4}$ is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ . Then use derivatives to argue the same thing.

Step by Step Solution

Step 1. Given information

Step 2. Using the definition

Step 3. Using the derivative

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Use the definitions of increasing and decreasing to argue that f(x)=x4 is decreasing on (−∞, 0] and increasing on [0,∞). Then use derivatives to argue the same thing.

Step by Step Solution

Step 1. Given information

Step 2. Using the definition

Step 3. Using the derivative

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Use the definitions of increasing and decreasing to argue that $f (x) = x^{4}$ is decreasing on $(- \infty, 0]$ and increasing on $[0, \infty)$ . Then use derivatives to argue the same thing.