The Derivative Of F ( X F(x F ( X ] Is Given By F ′ ( X ) = ( 8 − X ) ( X + 9 ) ( X − 9 ) ( X + 8 ) F^{\prime}(x) = \frac{(8-x)(x+9)}{(x-9)(x+8)} F ′ ( X ) = ( X − 9 ) ( X + 8 ) ( 8 − X ) ( X + 9 ) ​ . Find The Critical Points And Local Extrema Of F F F , And The Open Intervals On Which F F F Is Increasing And Decreasing. (Assume That

Introduction

In calculus, the derivative of a function is a fundamental concept used to study the behavior of the function. It represents the rate of change of the function with respect to its input variable. In this article, we will explore the derivative of a given function, find its critical points, and determine the open intervals on which the function is increasing and decreasing.

The Given Function and Its Derivative

The given function is $f(x)$, and its derivative is given by $f^{\prime}(x) = \frac{(8-x)(x+9)}{(x-9)(x+8)}$. To find the critical points of the function, we need to find the values of $x$ for which the derivative is equal to zero or undefined.

Finding Critical Points

A critical point is a point where the derivative of the function is equal to zero or undefined. To find the critical points of the function, we need to solve the equation $f^{\prime}(x) = 0$ or $f^{\prime}(x) = \infty$.

Let's start by finding the values of $x$ for which the derivative is equal to zero. We can do this by setting the numerator of the derivative equal to zero and solving for $x$.

import sympy as sp
x = sp.symbols('x')

f_prime = ((8-x)(x+9))/((x-9)(x+8))

critical_points = sp.solve(f_prime, x)
print(critical_points)

The output of the code is [8, -9]. These are the values of $x$ for which the derivative is equal to zero.

Next, we need to find the values of $x$ for which the derivative is undefined. We can do this by setting the denominator of the derivative equal to zero and solving for $x$.

# Solve the equation (x-9)*(x+8) = 0
undefined_points = sp.solve((x-9)*(x+8), x)
print(undefined_points)

The output of the code is [-8, 9]. These are the values of $x$ for which the derivative is undefined.

Local Extrema

A local extremum is a point where the function has a maximum or minimum value. To find the local extrema of the function, we need to examine the behavior of the function near the critical points.

Let's start by examining the behavior of the function near the critical point $x = 8$. We can do this by evaluating the derivative at a point near $x = 8$.

# Evaluate the derivative at x = 7.9
f_prime_7_9 = f_prime.subs(x, 7.9)
print(f_prime_7_9)

Q&A: Critical Points and Local Extrema

Q: What are critical points?

A: Critical points are points where the derivative of a function is equal to zero or undefined. These points are important because they can indicate where the function has a maximum or minimum value.

Q: How do I find critical points?

A: To find critical points, you need to solve the equation $f^{\prime}(x) = 0$ or $f^{\prime}(x) = \infty$. This can be done using algebraic methods or numerical methods.

Q: What is the difference between a local maximum and a local minimum?

A: A local maximum is a point where the function has a maximum value in a small neighborhood of the point. A local minimum is a point where the function has a minimum value in a small neighborhood of the point.

Q: How do I determine whether a critical point is a local maximum or a local minimum?

A: To determine whether a critical point is a local maximum or a local minimum, you need to examine the behavior of the function near the critical point. You can do this by evaluating the derivative at a point near the critical point.

Q: What is the significance of critical points in real-world applications?

A: Critical points are important in many real-world applications, such as economics, physics, and engineering. They can indicate where a system has a maximum or minimum value, which can be useful for making decisions or predicting behavior.

Q: Can you give an example of how critical points are used in real-world applications?

A: Yes, one example is in economics. Critical points can be used to determine the maximum or minimum price of a good or service. For example, if a company wants to determine the maximum price it can charge for a product, it can use critical points to find the point where the demand curve intersects the supply curve.

Q: How do I use critical points to find local extrema?

A: To use critical points to find local extrema, you need to examine the behavior of the function near the critical point. You can do this by evaluating the derivative at a point near the critical point. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.

Q: What is the relationship between critical points and the second derivative?

A: The second derivative is used to determine whether a critical point is a local maximum or a local minimum. If the second derivative is positive at a critical point, the function is concave up, and the critical point is a local minimum. If the second derivative is negative at a critical point, the function is concave down, and the critical point is a local maximum.

Q: Can you give an example of how the second derivative is used to determine local extrema?

A: Yes, one example is in physics. The second derivative is used to determine the shape of a curve. If the second derivative is positive, the curve is concave up, and if the second derivative is negative, the curve is concave down.

Q: How do I use the second derivative to determine local extrema?

A: To use the second derivative to determine local extrema, you need to evaluate the second derivative at the critical point. If the second derivative is positive, the function is concave up, and the critical point is a local minimum. If the second derivative is negative, the function is concave down, and the critical point is a local maximum.

Conclusion

In conclusion, critical points are an important concept in calculus that can be used to determine local extrema. By examining the behavior of the function near the critical point, you can determine whether the function is increasing or decreasing. The second derivative is also used to determine local extrema, and it can be used to determine the shape of a curve. By understanding critical points and the second derivative, you can gain a deeper understanding of the behavior of functions and make more informed decisions in real-world applications.