Find The Critical Numbers Of The Function F ( X ) = 12 X 5 + 15 X 4 − 20 X 3 − 1 F(x)=12x^5+15x^4-20x^3-1 F ( X ) = 12 X 5 + 15 X 4 − 20 X 3 − 1 And Classify Them Using A Graph.1. X = □ X = \square X = □ Is A □ \square □ (Select An Answer)2. X = □ X = \square X = □ Is A □ \square □ (Select An Answer)3.

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Introduction

In calculus, critical numbers are the values of x that make the derivative of a function equal to zero or undefined. These numbers are crucial in understanding the behavior of a function, as they can indicate the presence of local maxima, minima, or points of inflection. In this article, we will find the critical numbers of the function f(x)=12x5+15x4−20x3−1f(x)=12x^5+15x^4-20x^3-1 and classify them using a graph.

The Function and Its Derivative

The given function is f(x)=12x5+15x4−20x3−1f(x)=12x^5+15x^4-20x^3-1. To find the critical numbers, we need to find the derivative of this function.

f(x) = 12x^5 + 15x^4 - 20x^3 - 1
f'(x) = 60x^4 + 60x^3 - 60x^2

Finding Critical Numbers

Critical numbers are the values of x that make the derivative equal to zero or undefined. In this case, we need to find the values of x that make the derivative f′(x)f'(x) equal to zero.

f'(x) = 60x^4 + 60x^3 - 60x^2 = 0

To solve this equation, we can factor out the common term 60x260x^2.

60x^2(x^2 + x - 1) = 0

Now, we can set each factor equal to zero and solve for x.

60x^2 = 0 --> x = 0
x^2 + x - 1 = 0 --> x = -1 or x = 1

Therefore, the critical numbers of the function f(x)f(x) are x=−1,0,1x = -1, 0, 1.

Classifying Critical Numbers Using a Graph

To classify the critical numbers, we need to examine the behavior of the function around each critical number. We can do this by graphing the function and examining the sign of the derivative in each interval.

# Define the function
f(x) = 12x^5 + 15x^4 - 20x^3 - 1

f'(x) = 60x^4 + 60x^3 - 60x^2

Graphing the Function

To graph the function, we can use a graphing tool such as Desmos or a programming language such as Python.

import numpy as np
import matplotlib.pyplot as plt


def f(x):
return 12x**5 + 15x4 - 20*x3 - 1



def f_prime(x):
return 60x**4 + 60x3 - 60*x2



x = np.linspace(-2, 2, 400)



y = f(x)



plt.plot(x, y)
plt.title('Graph of f(x)')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.show()

Examining the Sign of the Derivative

To classify the critical numbers, we need to examine the sign of the derivative in each interval. We can do this by graphing the derivative and examining the sign of the derivative in each interval.

# Define the derivative
def f_prime(x):
    return 60*x**4 + 60*x**3 - 60*x**2


x = np.linspace(-2, 2, 400)



y = f_prime(x)



plt.plot(x, y)
plt.title('Graph of f'(x)')
plt.xlabel('x')
plt.ylabel('f'(x)')
plt.grid(True)
plt.show()

Classifying Critical Numbers

Based on the graph of the derivative, we can classify the critical numbers as follows:

  • x=−1x = -1 is a local maximum, as the derivative changes from positive to negative at this point.
  • x=0x = 0 is a local minimum, as the derivative changes from negative to positive at this point.
  • x=1x = 1 is a point of inflection, as the derivative changes from positive to negative at this point.

Conclusion

In this article, we found the critical numbers of the function f(x)=12x5+15x4−20x3−1f(x)=12x^5+15x^4-20x^3-1 and classified them using a graph. We used the derivative of the function to find the critical numbers and examined the behavior of the function around each critical number to classify them. The critical numbers were x=−1,0,1x = -1, 0, 1, and we classified them as a local maximum, a local minimum, and a point of inflection, respectively.

References

  • [1] Calculus: Early Transcendentals, 8th edition, by James Stewart.
  • [2] Graphing Calculators, by Texas Instruments.

Discussion

Introduction

In our previous article, we discussed finding critical numbers and classifying them using a graph. In this article, we will answer some frequently asked questions about critical numbers and graphing.

Q: What are critical numbers?

A: Critical numbers are the values of x that make the derivative of a function equal to zero or undefined. These numbers are crucial in understanding the behavior of a function, as they can indicate the presence of local maxima, minima, or points of inflection.

Q: How do I find critical numbers?

A: To find critical numbers, you need to find the values of x that make the derivative of a function equal to zero or undefined. You can do this by setting the derivative equal to zero and solving for x.

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

A: A local maximum is a point on a graph where the function changes from increasing to decreasing, while a local minimum is a point on a graph where the function changes from decreasing to increasing.

Q: How do I classify critical numbers using a graph?

A: To classify critical numbers using a graph, you need to examine the behavior of the function around each critical number. You can do this by graphing the function and examining the sign of the derivative in each interval.

Q: What is the second derivative, and how is it used in graphing?

A: The second derivative is the derivative of the derivative of a function. It is used to classify critical numbers and determine the concavity of a function.

Q: How do I use the second derivative to classify critical numbers?

A: To use the second derivative to classify critical numbers, you need to examine the sign of the second derivative at each critical number. If the second derivative is positive, the critical number is a local minimum. If the second derivative is negative, the critical number is a local maximum.

Q: What are some real-world applications of critical numbers?

A: Critical numbers have many real-world applications, including:

  • Optimization: Critical numbers are used to find the maximum or minimum of a function, which is essential in optimization problems.
  • Physics: Critical numbers are used to describe the motion of objects, such as the maximum height of a projectile.
  • Economics: Critical numbers are used to describe the behavior of economic systems, such as the maximum profit of a company.

Q: How do I graph a function using a graphing calculator?

A: To graph a function using a graphing calculator, you need to follow these steps:

  1. Enter the function into the calculator.
  2. Set the window to the desired range.
  3. Graph the function.

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

A: Some common mistakes to avoid when graphing include:

  • Incorrect window settings: Make sure the window is set to the correct range.
  • Incorrect function entry: Make sure the function is entered correctly.
  • Incorrect graphing: Make sure the graph is plotted correctly.

Conclusion

In this article, we answered some frequently asked questions about critical numbers and graphing. We discussed how to find critical numbers, classify them using a graph, and use the second derivative to classify critical numbers. We also discussed some real-world applications of critical numbers and provided tips for graphing using a graphing calculator.

References

  • [1] Calculus: Early Transcendentals, 8th edition, by James Stewart.
  • [2] Graphing Calculators, by Texas Instruments.

Discussion

What are some other questions you have about critical numbers and graphing? How can you apply critical numbers to real-world problems? What are some other ways to classify critical numbers?