Chapter 3: Q. 11 (page 260)

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.

Short Answer

Expert verified

The statement has been proven.

Step by step solution

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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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.

Short Answer

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.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Using the definition

Step 3. Using the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Calculus

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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.