The Function $f(x$\] Is Continuous Over The Interval \[-2,5\] And Has A Critical Point At $x=1$. If $f''(x$\] Is Negative On This Interval, Which Of The Following Is True?A. $f$ Has An Absolute Minimum At

Introduction

In calculus, a critical point is a point on the graph of a function where the derivative is zero or undefined. This concept is crucial in understanding the behavior of functions, particularly in optimization problems. In this article, we will explore the implications of a critical point on the continuity and differentiability of a function, specifically the function $f(x)$, which is continuous over the interval $[-2,5]$ and has a critical point at $x=1$. We will also examine the effect of a negative second derivative on the function's behavior.

Continuity and Differentiability

A function $f(x)$ is said to be continuous at a point $x=a$ if the following conditions are met:

1. The function is defined at $x=a$.
2. The limit of the function as $x$ approaches $a$ exists.
3. The limit of the function as $x$ approaches $a$ is equal to the value of the function at $x=a$.

On the other hand, a function $f(x)$ is said to be differentiable at a point $x=a$ if the derivative of the function exists at that point. The derivative of a function $f(x)$ is denoted by $f'(x)$ and represents the rate of change of the function with respect to $x$.

Critical Points

A critical point of a function $f(x)$ is a point where the derivative $f'(x)$ is zero or undefined. In the case of the function $f(x)$, we are given that it has a critical point at $x=1$. This means that the derivative $f'(x)$ is either zero or undefined at $x=1$.

Second Derivative

The second derivative of a function $f(x)$ is denoted by $f''(x)$ and represents the rate of change of the derivative with respect to $x$. In other words, the second derivative measures how quickly the rate of change of the function is changing.

Negative Second Derivative

If the second derivative $f''(x)$ is negative on the interval $[-2,5]$, it means that the rate of change of the derivative is decreasing. This implies that the function is concave down, meaning that it is curving downward.

Implications of a Negative Second Derivative

A negative second derivative has several implications on the behavior of the function. Some of these implications include:

• The function is concave down, meaning that it is curving downward.
• The function has a local maximum at the critical point $x=1$.
• The function has an absolute minimum at the endpoint $x=5$.

Conclusion

In conclusion, the function $f(x)$, which is continuous over the interval $[-2,5]$ and has a critical point at $x=1$, has a negative second derivative on this interval. This implies that the function is concave down, has a local maximum at the critical point $x=1$, and has an absolute minimum at the endpoint $x=5$.

Discussion

The implications of a negative second derivative on the behavior of a function are crucial in understanding the behavior of functions, particularly in optimization problems. By analyzing the second derivative, we can determine the concavity of the function and identify local maxima and minima.

References

• [1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Pearson Education.
• [2] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2013). Calculus: A New Horizon. John Wiley & Sons.

Final Answer

The final answer is: $\boxed{C}$

Introduction

In our previous article, we explored the implications of a critical point on the continuity and differentiability of a function, specifically the function $f(x)$, which is continuous over the interval $[-2,5]$ and has a critical point at $x=1$. We also examined the effect of a negative second derivative on the function's behavior. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is a critical point, and how is it related to the derivative of a function?

A1: A critical point is a point on the graph of a function where the derivative is zero or undefined. This means that the derivative of the function is either zero or undefined at that point.

Q2: How does a negative second derivative affect the behavior of a function?

A2: A negative second derivative implies that the function is concave down, meaning that it is curving downward. This also means that the function has a local maximum at the critical point and an absolute minimum at the endpoint.

Q3: What is the significance of a critical point in optimization problems?

A3: A critical point is significant in optimization problems because it represents a point where the function may have a maximum or minimum value. By analyzing the second derivative at the critical point, we can determine whether it is a local maximum or minimum.

Q4: How can we determine the concavity of a function using the second derivative?

A4: We can determine the concavity of a function by analyzing the sign of the second derivative. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down.

Q5: What is the relationship between the second derivative and the first derivative?

A5: The second derivative is the derivative of the first derivative. In other words, it measures how quickly the rate of change of the function is changing.

Q6: Can a function have multiple critical points?

A6: Yes, a function can have multiple critical points. Each critical point represents a point where the derivative is zero or undefined.

Q7: How can we use the second derivative to identify local maxima and minima?

A7: We can use the second derivative to identify local maxima and minima by analyzing the sign of the second derivative at the critical point. If the second derivative is negative, the function has a local maximum at the critical point, and if it is positive, the function has a local minimum.

Q8: What is the significance of the interval $[-2,5]$ in this problem?

A8: The interval $[-2,5]$ is significant in this problem because it represents the domain of the function $f(x)$. The function is continuous over this interval, and the critical point is located at $x=1$.

Conclusion

In conclusion, the function $f(x)$, which is continuous over the interval $[-2,5]$ and has a critical point at $x=1$, has a negative second derivative on this interval. This implies that the function is concave down, has a local maximum at the critical point $x=1$, and has an absolute minimum at the endpoint $x=5$. We hope that this Q&A article has provided a better understanding of the implications of a critical point on the continuity and differentiability of a function.

Discussion

The implications of a critical point on the continuity and differentiability of a function are crucial in understanding the behavior of functions, particularly in optimization problems. By analyzing the second derivative, we can determine the concavity of the function and identify local maxima and minima.

References

• [1] Thomas, G. B. (2010). Calculus and Analytic Geometry. Pearson Education.
• [2] Stewart, J. (2012). Calculus: Early Transcendentals. Cengage Learning.
• [3] Anton, H. (2013). Calculus: A New Horizon. John Wiley & Sons.

Final Answer

The final answer is: $\boxed{C}$