Problem 6 Plot the graph of the function \... [FREE SOLUTION]

Chapter 0: Problem 6

Plot the graph of the function $f$ in an appropriate viewing window. (Note: The answer is not unique.) $$ \begin{array}{l} f(x)=-2 x^{4}+5 x^{2}-4\\\ \text { 7. } f(x)=\frac{x^{3}}{x^{3}+1} \end{array} $$

Short Answer

Expert verified

Considering the critical points at $x = 0$ and $x = \pm \frac{\sqrt{5}}{2}$, and the end behavior as $x \to \infty$ and $x \to -\infty$, an appropriate viewing window for the function $f(x) = -2x^4 + 5x^2 - 4$ could be given by: - For x-axis: From $-3$ to $3$ - For y-axis: From $-10$ to $5$ Note that this is just one possible viewing window and there can be other windows that would still display important features of the graph.

Step by step solution

Determine the domain and range of the function

Since the function is a polynomial, its domain is all real numbers ($x \in \mathbb{R}$). The range of the function will be determined after finding critical points and end behavior.

Calculate the first derivative

To find the critical points of the function, we must first calculate its first derivative. \[ f'(x) = \frac{d}{dx} (-2x^4 + 5x^2 - 4) = -8x^3 + 10x \]

Find critical points

Now, find the critical points by setting the first derivative equal to zero and solve for $x$: \[ -8x^3 + 10x = 0 \] Factoring, we get: \[ -2x(4x^2 - 5) = 0 \] So the critical points are: $x = 0$ and $x = \pm \frac{\sqrt{5}}{2}$.

Analyze end behavior

As $x$ approaches infinity or negative infinity, the term with the highest degree ($-2x^4$) will dominate the polynomial, making the function's end behavior: \[ \lim_{x \to \infty} f(x) = -\infty \] \[ \lim_{x \to -\infty} f(x) = -\infty \]

Determine the appropriate viewing window

Considering the critical points and the end behavior, a good viewing window for the function would incorporate the following limits: - For x-axis: From $-3$ to $3$, which will capture the critical points and a bit beyond. - For y-axis: From $-10$ to $5$, which will include the end behavior and the global minimum which lies somewhere near to the lower range. Plot the graph of $f(x) = -2x^4 + 5x^2 - 4$ using the viewing window with the above limits. Note that this is just one possible viewing window and there can be other windows that would still display important features of the graph.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ \begin{array}{l} f(x)=-2 x^{4}+5 x^{2}-4\\\ \text { 7. } f(x)=\frac{x^{3}}{x^{3}+1} \end{array} $$

Short Answer

Step by step solution

Determine the domain and range of the function

Calculate the first derivative

Find critical points

Analyze end behavior

Determine the appropriate viewing window

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Calculus

Probability and Statistics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.