Chapter 4: Q. 84 (page 210)

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} + 12 x^{4} - x - 3$

Short Answer

Expert verified

For the given polynomial function, we get:

x-intercept $= - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, - 3$

y-intercept $= - 3$

Step by step solution

Step 1. End behavior of the function

We have a degree of five from $f (x) = 4 x^{5} + 12 x^{4} - x - 3$ .

Thus, the graph of function has an end behavior similar to $y = 4 x^{5}$ .

Step 2. Determine the intercept of the function

Determine the intercept of function to get the x- intercept.

$\begin{aligned} 4 x^{5} + 12 x^{4} - x - 3 = 0 \\ 4 x^{4} (x + 3) - (x + 3) = 0 \\ (4 x^{4} - 1) (x + 3) = 0 \\ 2 x^{2} + 1 = 0; \begin{aligned} (2 x^{2} + 1) (2 x^{2} - 1) (x + 3) & = 0 \\ 2 x^{2} - 1 & = 0; x + 3 & = 0 \\ 2 x^{2} & = - 1; & 2 x^{2} & = 1; & x & = - 3 \\ \sqrt{x^{2}} & = - \sqrt{\frac{1}{2};} & \sqrt{x^{2}} & = \sqrt{\frac{1}{2}} \\ x & = - \frac{\sqrt{2}}{2}; & x & = \frac{\sqrt{2}}{2} \end{aligned} \end{aligned}$

Considering $f (0) = - 3$ we have y-intercept $(0, - 3)$

Step 3. Graph the function

Step 4. Maximum and minimum turning point

From the graph above, we can see that the maximum and minimum turning point of the function is:

$(- 2.404, 79.028)$ and $(0.266, - 3.201)$

Step 5. Graph of function with obtained result

Step 6. Domain and range

Therefore, the domain and range of the function is the set of all real numbers.

Step 7. Interval at which the graph is increasing or decreasing

By observing the graph above, we say that the function at the intervals of $(- \infty, 2.404)$ and $(0.266, \infty)$ is increasing.

At intervals of $(- 2.404, 0.266)$ decreasing.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} + 12 x^{4} - x - 3$

Short Answer

Step by step solution

Step 1. End behavior of the function

Step 2. Determine the intercept of the function

Step 3. Graph the function

Step 4. Maximum and minimum turning point

Step 5. Graph of function with obtained result

Step 6. Domain and range

Step 7. Interval at which the graph is increasing or decreasing

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Statistics

Calculus

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1f(x)=4x5+12x4-x-3

Short Answer

Step by step solution

Step 1. End behavior of the function

Step 2. Determine the intercept of the function

Step 3. Graph the function

Step 4. Maximum and minimum turning point

Step 5. Graph of function with obtained result

Step 6. Domain and range

Step 7. Interval at which the graph is increasing or decreasing

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Statistics

Calculus

Mechanics Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} + 12 x^{4} - x - 3$