Critical Numbers Of F(x) = 6x⁵ + 15x⁴ - 20x³ - 7 Graphical Classification

Introduction

In calculus, identifying and classifying critical numbers of a function is a fundamental step in understanding its behavior. Critical numbers are the points where the derivative of the function is either zero or undefined. These points are crucial because they can indicate local maxima, local minima, or saddle points, which are essential for sketching the graph of the function and solving optimization problems. This article aims to guide you through the process of finding the critical numbers of the function f(x) = 6x⁵ + 15x⁴ - 20x³ - 7 and classifying them using its graph. We will delve into the necessary calculus techniques, including differentiation and graphical analysis, to provide a comprehensive understanding of the function's critical points.

The process involves several key steps, starting with finding the first derivative of the function. This derivative will then be used to identify the critical points by setting it equal to zero and solving for x. Once the critical numbers are found, we will analyze the second derivative or use a sign chart to determine whether each critical point corresponds to a local maximum, a local minimum, or neither. Finally, we will use a graphical approach to visually confirm our findings and gain a deeper insight into the function's behavior. This method not only helps in classifying the critical points but also enhances our understanding of how the function changes its direction and concavity across its domain.

Understanding critical numbers is vital in various fields beyond mathematics, including physics, engineering, economics, and computer science. For instance, in physics, these points can represent equilibrium states of a system, while in economics, they can indicate points of maximum profit or minimum cost. Therefore, a thorough grasp of how to find and interpret critical numbers is invaluable for anyone dealing with mathematical modeling and optimization in real-world applications. This article provides a detailed, step-by-step guide that will equip you with the necessary skills to analyze and classify critical points effectively.

Step 1: Find the First Derivative

To begin, we need to find the first derivative of the function f(x) = 6x⁵ + 15x⁴ - 20x³ - 7. The derivative, denoted as f'(x), will help us identify the critical points where the function's slope is either zero or undefined. Using the power rule for differentiation, which states that the derivative of xⁿ is nxⁿ⁻¹, we can differentiate each term of the function separately. This process involves applying the power rule and constant multiple rule to each term of the original function, ensuring that we accurately capture the rate of change at any point along the curve.

The power rule is a cornerstone of differential calculus, and its correct application is crucial for finding derivatives. For our function, each term is a power of x multiplied by a constant, making the power rule directly applicable. We start by multiplying the exponent of x by the coefficient of the term and then reduce the exponent by one. For example, the derivative of 6x⁵ is 30x⁴, obtained by multiplying 6 by 5 and reducing the exponent from 5 to 4. This process is repeated for each term in the function, ensuring we account for all parts of the polynomial.

Applying the power rule to each term, we get:

• The derivative of 6x⁵ is 30x⁴.
• The derivative of 15x⁴ is 60x³.
• The derivative of -20x³ is -60x².
• The derivative of -7 (a constant) is 0.

Adding these derivatives together, we find the first derivative of f(x):

f'(x) = 30x⁴ + 60x³ - 60x²

This first derivative, f'(x), is a crucial tool for finding the critical numbers. It represents the slope of the tangent line to the function f(x) at any point x. Critical points occur where this slope is either zero or undefined. In the next step, we will set f'(x) equal to zero and solve for x to find these critical numbers. Understanding the derivative as the slope of the tangent line helps us visualize the points where the function may change direction, which are precisely the points we are trying to identify.

Step 2: Find the Critical Numbers

Now that we have the first derivative, f'(x) = 30x⁴ + 60x³ - 60x², the next step is to find the critical numbers. Critical numbers are the values of x for which f'(x) = 0 or f'(x) is undefined. In this case, f'(x) is a polynomial, so it is defined for all real numbers. Thus, we only need to find the values of x for which f'(x) = 0. This involves solving the equation 30x⁴ + 60x³ - 60x² = 0 for x. The solutions to this equation will be the critical points of the function, which are potential locations of local maxima, local minima, or saddle points.

To solve the equation, we first look for common factors that can be factored out. Observing the terms in the derivative, we can see that 30x² is a common factor in all terms. Factoring out 30x² simplifies the equation and makes it easier to find the roots. This step is crucial because it reduces the degree of the polynomial we need to solve, making the process more manageable. Factoring out common terms is a standard technique in algebra and is particularly useful when dealing with polynomial equations.

Factoring out 30x² from 30x⁴ + 60x³ - 60x² = 0, we get:

30x²(x² + 2x - 2) = 0

This factored form of the equation now allows us to use the zero-product property, which states that if the product of several factors is zero, then at least one of the factors must be zero. Thus, we set each factor equal to zero and solve for x. The first factor, 30x², gives us the solution x = 0. The second factor, x² + 2x - 2, is a quadratic equation, which we can solve using the quadratic formula.

The quadratic formula is a general method for solving quadratic equations of the form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, x² + 2x - 2 = 0, we have a = 1, b = 2, and c = -2. Plugging these values into the quadratic formula, we get:

x = (-2 ± √(2² - 4(1)(-2))) / (2(1))
x = (-2 ± √(4 + 8)) / 2
x = (-2 ± √12) / 2
x = (-2 ± 2√3) / 2
x = -1 ± √3

Thus, the solutions from the quadratic factor are x = -1 + √3 and x = -1 - √3. Combining these with the solution from the first factor, x = 0, we have three critical numbers:

• x = 0
• x = -1 + √3
• x = -1 - √3

These critical numbers are the points where the function's slope may change direction, indicating potential local maxima, local minima, or saddle points. In the next step, we will use a graphical approach to classify these critical points.

Step 3: Classify Critical Numbers Using a Graph

After finding the critical numbers, the next crucial step is to classify them. Classification involves determining whether each critical number corresponds to a local maximum, a local minimum, or neither. One effective method to classify these points is by analyzing the graph of the function. A graph provides a visual representation of the function's behavior, making it easier to identify where the function changes direction and to classify the critical points accordingly. This approach is particularly useful for understanding the overall shape of the function and how it behaves around its critical points.

To classify the critical numbers using a graph, we can either sketch the graph by hand or use a graphing calculator or software. For accuracy and ease, using graphing software like Desmos or Wolfram Alpha is highly recommended. These tools allow you to plot the function f(x) = 6x⁵ + 15x⁴ - 20x³ - 7 and visually inspect its behavior around the critical points. The graph will show the function's curve, its peaks, and its valleys, which directly correspond to local maxima and minima.

First, let's approximate the values of the critical numbers we found earlier:

• x = 0
• x = -1 + √3 ≈ 0.732
• x = -1 - √3 ≈ -2.732

These are the x-coordinates where we expect to see significant changes in the function's behavior. When we plot the function, we will focus on the intervals around these points to determine the nature of the critical points. The key is to observe how the function's value changes as x passes through each critical number.

When examining the graph, we look for the following:

• Local Maximum: A critical point where the function changes from increasing to decreasing. On the graph, this appears as a peak.
• Local Minimum: A critical point where the function changes from decreasing to increasing. On the graph, this appears as a valley.
• Saddle Point (or Neither): A critical point where the function does not change direction. The function might flatten out at this point but continues to either increase or decrease. This point is neither a maximum nor a minimum.

By plotting the function f(x) = 6x⁵ + 15x⁴ - 20x³ - 7, we can observe the following behavior:

• At x ≈ -2.732, the function changes from decreasing to increasing. This indicates a local minimum.
• At x = 0, the function flattens out but continues to decrease on both sides. This indicates a saddle point (or neither a local maximum nor a local minimum).
• At x ≈ 0.732, the function changes from increasing to decreasing. This indicates a local maximum.

Thus, using the graph, we have classified the critical numbers as follows:

• x = -1 - √3 is a local minimum.
• x = 0 is a saddle point.
• x = -1 + √3 is a local maximum.

This graphical classification provides a clear and intuitive understanding of the function's behavior around its critical points. It complements the analytical methods of finding critical numbers and helps in sketching an accurate representation of the function. In the next section, we will summarize our findings and reinforce the key steps in the process.

Conclusion

In this article, we successfully found and classified the critical numbers of the function f(x) = 6x⁵ + 15x⁴ - 20x³ - 7. The process involved several key steps, each contributing to a comprehensive understanding of the function's behavior. We began by finding the first derivative of the function, which is a fundamental step in identifying critical points. This derivative, f'(x) = 30x⁴ + 60x³ - 60x², represents the slope of the tangent line to the function at any point and is crucial for locating where the function may change direction.

Next, we identified the critical numbers by setting the first derivative equal to zero and solving for x. This led us to the equation 30x⁴ + 60x³ - 60x² = 0, which we simplified by factoring out the common term 30x². The resulting equation, 30x²(x² + 2x - 2) = 0, provided us with one critical number directly, x = 0, and a quadratic equation to solve. Using the quadratic formula, we found the other two critical numbers, x = -1 + √3 and x = -1 - √3. These critical numbers are potential locations for local maxima, local minima, or saddle points.

To classify these critical numbers, we employed a graphical approach. By plotting the function using graphing software, we visually analyzed the function's behavior around each critical point. This method allowed us to determine whether the function changed from increasing to decreasing (indicating a local maximum), from decreasing to increasing (indicating a local minimum), or neither (indicating a saddle point). The graphical analysis revealed that x = -1 - √3 is a local minimum, x = 0 is a saddle point, and x = -1 + √3 is a local maximum.

The combination of analytical and graphical methods provides a robust approach to finding and classifying critical numbers. This process not only helps in understanding the specific function at hand but also reinforces fundamental calculus concepts and techniques. The ability to find and classify critical numbers is invaluable in various fields, including optimization problems in engineering, economics, and computer science. By mastering these techniques, one can gain deeper insights into the behavior of functions and their applications in real-world scenarios.

In summary, this article has provided a step-by-step guide to finding and classifying critical numbers using both analytical and graphical methods. The understanding gained from this process is essential for anyone working with mathematical functions and their applications, making it a crucial topic in calculus and beyond.