Which Of The Following Graphs Could Be The Graph Of The Function $f(x) = X^4 + X^3 - X^2 - X$?

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Introduction to Graphing Polynomial Functions

Graphing polynomial functions can be a complex task, especially when dealing with high-degree polynomials. However, by understanding the properties of polynomial functions and using various techniques, we can determine the possible graphs of a given function. In this article, we will explore the graph of the function $f(x) = x^4 + x^3 - x^2 - x$ and determine which of the given graphs could be its representation.

Understanding the Function

The given function is a fourth-degree polynomial, which means it can have at most four turning points. The function can be written as:

$$f(x) = x^4 + x^3 - x^2 - x$$

To understand the behavior of this function, we can start by finding its first and second derivatives.

Finding the First and Second Derivatives

The first derivative of the function is:

$$f'(x) = 4x^3 + 3x^2 - 2x - 1$$

The second derivative of the function is:

$$f''(x) = 12x^2 + 6x - 2$$

Analyzing the First and Second Derivatives

By analyzing the first and second derivatives, we can determine the critical points and inflection points of the function. Critical points occur where the first derivative is equal to zero, and inflection points occur where the second derivative is equal to zero.

Finding Critical Points

To find the critical points, we set the first derivative equal to zero and solve for x:

$$4x^3 + 3x^2 - 2x - 1 = 0$$

Solving this equation, we find that there are three real roots: $x = -1$, $x = 0$, and $x = 1$.

Finding Inflection Points

To find the inflection points, we set the second derivative equal to zero and solve for x:

$$12x^2 + 6x - 2 = 0$$

Solving this equation, we find that there are two real roots: $x = -\frac{1}{2}$ and $x = \frac{1}{3}$.

Understanding the Graph

Now that we have found the critical points and inflection points, we can understand the behavior of the function. The function has three real roots, which means it crosses the x-axis at three points. The function also has two inflection points, which means it changes concavity at these points.

Possible Graphs

Based on the analysis above, we can determine the possible graphs of the function. The graph must have three real roots, two inflection points, and a concavity change at each inflection point.

Graph 1: A Graph with Three Real Roots and Two Inflection Points

This graph has three real roots and two inflection points, which matches the analysis above. The graph also has a concavity change at each inflection point, which is consistent with the second derivative.

Graph 2: A Graph with Four Real Roots and Two Inflection Points

This graph has four real roots, which is not consistent with the analysis above. The graph also has two inflection points, but the concavity change at each inflection point is not consistent with the second derivative.

Graph 3: A Graph with Three Real Roots and One Inflection Point

This graph has three real roots, but only one inflection point, which is not consistent with the analysis above. The graph also does not have a concavity change at the inflection point, which is not consistent with the second derivative.

Conclusion

Based on the analysis above, we can conclude that the graph of the function $f(x) = x^4 + x^3 - x^2 - x$ must have three real roots, two inflection points, and a concavity change at each inflection point. The only graph that matches this description is Graph 1.

Final Answer

The final answer is Graph 1.

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Introduction

In our previous article, we explored the graph of the function $f(x) = x^4 + x^3 - x^2 - x$ and determined that the possible graph must have three real roots, two inflection points, and a concavity change at each inflection point. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What are the critical points of the function?

A: The critical points of the function are the values of x where the first derivative is equal to zero. In this case, the critical points are $x = -1$, $x = 0$, and $x = 1$.

Q: What are the inflection points of the function?

A: The inflection points of the function are the values of x where the second derivative is equal to zero. In this case, the inflection points are $x = -\frac{1}{2}$ and $x = \frac{1}{3}$.

Q: What is the concavity of the function at each inflection point?

A: The concavity of the function at each inflection point can be determined by analyzing the second derivative. At $x = -\frac{1}{2}$, the second derivative is positive, indicating that the function is concave up. At $x = \frac{1}{3}$, the second derivative is negative, indicating that the function is concave down.

Q: How many real roots does the function have?

A: The function has three real roots, which means it crosses the x-axis at three points.

Q: What is the degree of the polynomial function?

A: The degree of the polynomial function is 4, which means it can have at most four turning points.

Q: How can I determine the possible graphs of a polynomial function?

A: To determine the possible graphs of a polynomial function, you can analyze the first and second derivatives of the function. Critical points occur where the first derivative is equal to zero, and inflection points occur where the second derivative is equal to zero.

Q: What is the significance of the inflection points in the graph of the function?

A: The inflection points in the graph of the function indicate where the concavity of the function changes. This can provide valuable information about the behavior of the function.

Q: How can I use the graph of the function to make conclusions about its behavior?

A: You can use the graph of the function to make conclusions about its behavior by analyzing the critical points, inflection points, and concavity of the function.

Q: What are some common mistakes to avoid when graphing polynomial functions?

A: Some common mistakes to avoid when graphing polynomial functions include:

• Not analyzing the first and second derivatives of the function
• Not identifying the critical points and inflection points of the function
• Not considering the concavity of the function at each inflection point

Conclusion

In this article, we have answered some frequently asked questions about the graph of the function $f(x) = x^4 + x^3 - x^2 - x$. We have discussed the critical points, inflection points, and concavity of the function, and provided tips on how to determine the possible graphs of a polynomial function.

Final Answer

The final answer is that the graph of the function $f(x) = x^4 + x^3 - x^2 - x$ must have three real roots, two inflection points, and a concavity change at each inflection point.