Select The Correct Answer.Consider This Function:${ F(x) = 6 \log_2 X - 3 }$Over Which Interval Is Function { F $}$ Increasing At The Greatest Rate?A. { [2, 6]$}$ B. { \left[\frac{1}{8}, \frac{1}{2}\right]$}$

Introduction

In calculus, the rate of change of a function is a fundamental concept that helps us understand how the function behaves over different intervals. When a function is increasing at the greatest rate, it means that the function is not only increasing but also doing so at the fastest possible pace. In this article, we will explore how to determine the interval over which a given function is increasing at the greatest rate.

The Given Function

The function we are given is:

${ f(x) = 6 \log_2 x - 3 }$

This is a logarithmic function with base 2, and it has a linear coefficient of 6. The constant term is -3.

Understanding the Rate of Change

To determine the rate of change of a function, we need to find its derivative. The derivative of a function represents the rate of change of the function with respect to its input variable.

Finding the Derivative

To find the derivative of the given function, we will use the chain rule and the fact that the derivative of $\log_b x$ is $\frac{1}{x \ln b}$.

${ f'(x) = 6 \cdot \frac{1}{x \ln 2} }$

Simplifying the expression, we get:

${ f'(x) = \frac{6}{x \ln 2} }$

Determining the Interval of Maximum Rate of Increase

To determine the interval over which the function is increasing at the greatest rate, we need to find the critical points of the function. Critical points occur when the derivative of the function is equal to zero or undefined.

In this case, the derivative is undefined when $x = 0$, since the logarithm of zero is undefined. Therefore, the critical point is $x = 0$.

However, we are interested in finding the interval over which the function is increasing at the greatest rate. To do this, we need to find the intervals over which the derivative is positive.

Analyzing the Sign of the Derivative

The derivative is positive when $\frac{6}{x \ln 2} > 0$. Since the numerator is positive, the sign of the derivative depends on the denominator.

The denominator is positive when $x > 0$. Therefore, the derivative is positive when $x > 0$.

Finding the Maximum Rate of Increase

To find the maximum rate of increase, we need to find the point at which the derivative is maximum. This occurs when the derivative is equal to its maximum value.

Since the derivative is positive for all $x > 0$, the maximum rate of increase occurs when the derivative is maximum.

Comparing the Options

We are given two options:

A. $[2, 6]$ B. $\left[\frac{1}{8}, \frac{1}{2}\right]$

To determine which option is correct, we need to compare the values of the derivative at the endpoints of each interval.

Evaluating the Derivative at the Endpoints

Let's evaluate the derivative at the endpoints of each interval:

For option A:

• At $x = 2$, the derivative is $\frac{6}{2 \ln 2} = \frac{3}{\ln 2}$
• At $x = 6$, the derivative is $\frac{6}{6 \ln 2} = \frac{1}{\ln 2}$

For option B:

• At $x = \frac{1}{8}$, the derivative is $\frac{6}{\frac{1}{8} \ln 2} = 48 \ln 2$
• At $x = \frac{1}{2}$, the derivative is $\frac{6}{\frac{1}{2} \ln 2} = 12 \ln 2$

Comparing the Values

Comparing the values of the derivative at the endpoints, we can see that the derivative is maximum at $x = \frac{1}{8}$.

Therefore, the correct option is:

B. $\left[\frac{1}{8}, \frac{1}{2}\right]$

Conclusion

In this article, we have determined the interval over which the given function is increasing at the greatest rate. We have found that the maximum rate of increase occurs when the derivative is maximum, and we have compared the values of the derivative at the endpoints of each interval to determine the correct option.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Discussion

Introduction

In our previous article, we explored how to determine the interval over which a given function is increasing at the greatest rate. We used the concept of rate of change and the derivative of a function to find the maximum rate of increase.

In this article, we will answer some frequently asked questions related to determining the interval of maximum rate of increase for a logarithmic function.

Q: What is the rate of change of a function?

A: The rate of change of a function is a measure of how fast the function is changing with respect to its input variable. It is represented by the derivative of the function.

Q: How do I find the derivative of a logarithmic function?

A: To find the derivative of a logarithmic function, you can use the chain rule and the fact that the derivative of $\log_b x$ is $\frac{1}{x \ln b}$.

Q: What is the critical point of a function?

A: The critical point of a function is a point at which the derivative of the function is equal to zero or undefined. In the case of a logarithmic function, the critical point is the point at which the function is undefined, which is $x = 0$.

Q: How do I determine the interval over which a function is increasing at the greatest rate?

A: To determine the interval over which a function is increasing at the greatest rate, you need to find the critical points of the function and compare the values of the derivative at the endpoints of each interval.

Q: What is the maximum rate of increase of a function?

A: The maximum rate of increase of a function is the point at which the derivative of the function is maximum. This occurs when the derivative is equal to its maximum value.

Q: How do I compare the values of the derivative at the endpoints of each interval?

A: To compare the values of the derivative at the endpoints of each interval, you need to evaluate the derivative at each endpoint and compare the values.

Q: What is the correct option for the interval over which the function is increasing at the greatest rate?

A: Based on our previous article, the correct option is:

B. $\left[\frac{1}{8}, \frac{1}{2}\right]$

Q: Can you provide more examples of how to determine the interval of maximum rate of increase for a logarithmic function?

A: Yes, here are a few more examples:

• Example 1: Find the interval over which the function $f(x) = 3 \log_2 x + 2$ is increasing at the greatest rate.
• Example 2: Find the interval over which the function $f(x) = 2 \log_3 x - 1$ is increasing at the greatest rate.

Conclusion

In this article, we have answered some frequently asked questions related to determining the interval of maximum rate of increase for a logarithmic function. We have provided examples and explanations to help you understand the concept.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Discussion

Do you have any questions or topics you would like to discuss related to determining the interval of maximum rate of increase for a logarithmic function? Share your thoughts and ideas in the comments below!