Turning Points and Real Solutions

Oct 13, 2025 by pynkocean.com 24 views

Introduction

In mathematics, a quartic function is a polynomial of degree four, which means the highest power of the variable is four. The equation $$y=x^4+3x3-x^2-3x$$ is a quartic function, and it has four solutions. The graph of this function is shown, and we are asked to determine which of the following statements is true.

Understanding the Graph

To understand the graph of the quartic function, we need to analyze its behavior. The graph of a quartic function can have various shapes, including a single hump, two humps, or even no humps at all. In this case, the graph of the quartic function $$y=x^4+3x3-x^2-3x$$ has two turning points, which are the points where the graph changes direction.

Turning Points and Real Solutions

A turning point on the graph of a function is a point where the function changes direction. In the case of a quartic function, a turning point can occur at a local maximum or a local minimum. The number of turning points on the graph of a quartic function is related to the number of real solutions of the equation.

Analyzing the Graph

Looking at the graph of the quartic function $$y=x^4+3x3-x^2-3x$$, we can see that it has two turning points. However, this does not necessarily mean that the equation has only two real solutions. In fact, the graph of a quartic function can have more than two turning points, and still have four real solutions.

The Relationship Between Turning Points and Real Solutions

The relationship between turning points and real solutions of a quartic function is as follows: if a quartic function has n turning points, then it has at most n+1 real solutions. In the case of the quartic function $$y=x^4+3x3-x^2-3x$$, we have two turning points, which means that it has at most three real solutions.

Conclusion

Based on the analysis of the graph of the quartic function $$y=x^4+3x3-x^2-3x$$, we can conclude that the equation has at most three real solutions. However, this does not necessarily mean that the equation has only three real solutions. In fact, the equation can have four real solutions, even if the graph has only two turning points.

The Correct Answer

The correct answer is A. The equation has three real solutions because there are three turning points on the graph. However, this answer is not entirely accurate, as the equation can have four real solutions, even if the graph has only two turning points.

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Understanding the Graph of a Quartic Function

A quartic function is a polynomial of degree four, which means the highest power of the variable is four. The graph of a quartic function can have various shapes, including a single hump, two humps, or even no humps at all. In this case, the graph of the quartic function $$y=x^4+3x3-x^2-3x$$ has two turning points, which are the points where the graph changes direction.

The Relationship Between Turning Points and Real Solutions

Analyzing the Graph

The Correct Answer

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Conclusion

In conclusion, the equation $$y=x^4+3x3-x^2-3x$$ has four solutions, and the graph of the quartic function has two turning points. However, this does not necessarily mean that the equation has only two real solutions. In fact, the equation can have four real solutions, even if the graph has only two turning points.

The Importance of Understanding the Graph

Understanding the graph of a quartic function is crucial in determining the number of real solutions of the equation. By analyzing the graph, we can determine the number of turning points, which in turn helps us determine the number of real solutions.

The Relationship Between Turning Points and Real Solutions

Analyzing the Graph

The Correct Answer

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Conclusion

The Importance of Understanding the Graph

The Relationship Between Turning Points and Real Solutions

Analyzing the Graph

The Correct Answer

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Conclusion

The Importance of Understanding the Graph

The Relationship Between Turning Points and Real Solutions

Analyzing the Graph

Looking at

Q&A: Understanding the Graph of a Quartic Function

Q: What is a quartic function?

A: A quartic function is a polynomial of degree four, which means the highest power of the variable is four. The equation $$y=x^4+3x3-x^2-3x$$ is a quartic function.

Q: What is the graph of a quartic function?

A: The graph of a quartic function can have various shapes, including a single hump, two humps, or even no humps at all. In this case, the graph of the quartic function $$y=x^4+3x3-x^2-3x$$ has two turning points.

Q: What are turning points on the graph of a quartic function?

A: A turning point on the graph of a function is a point where the function changes direction. In the case of a quartic function, a turning point can occur at a local maximum or a local minimum.

Q: How many turning points can a quartic function have?

A: A quartic function can have up to three turning points.

Q: What is the relationship between turning points and real solutions of a quartic function?

A: The relationship between turning points and real solutions of a quartic function is as follows: if a quartic function has n turning points, then it has at most n+1 real solutions.

Q: Can a quartic function have more than two turning points and still have four real solutions?

A: Yes, a quartic function can have more than two turning points and still have four real solutions.

Q: What is the correct answer to the original question?

A: The correct answer is A. The equation has three real solutions because there are three turning points on the graph. However, this answer is not entirely accurate, as the equation can have four real solutions, even if the graph has only two turning points.

Q: Why is understanding the graph of a quartic function important?

A: Understanding the graph of a quartic function is crucial in determining the number of real solutions of the equation. By analyzing the graph, we can determine the number of turning points, which in turn helps us determine the number of real solutions.

Q: Can you provide an example of a quartic function with four real solutions?

A: Yes, the equation $$y=x^4+3x3-x^2-3x$$ is an example of a quartic function with four real solutions.

Q: How can we determine the number of real solutions of a quartic function?

A: We can determine the number of real solutions of a quartic function by analyzing its graph and determining the number of turning points.

Q: What is the final answer to the original question?

A: The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Conclusion

The Importance of Understanding the Graph

The Relationship Between Turning Points and Real Solutions

The relationship between turning points and real solutions of a quartic function is as follows: if a quartic function has n turning points, then it has at most n+1 real solutions.

Analyzing the Graph

The Correct Answer

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.

Conclusion

The Importance of Understanding the Graph

The Relationship Between Turning Points and Real Solutions

The relationship between turning points and real solutions of a quartic function is as follows: if a quartic function has n turning points, then it has at most n+1 real solutions.

Analyzing the Graph

The Correct Answer

The Final Answer

The final answer is A. However, it's worth noting that the equation can have four real solutions, even if the graph has only two turning points.