Absolute Extrema Of F(x) = 3x^4 - 2x^3 On [-4, 4]

Introduction to Extreme Value Analysis

In the realm of mathematical analysis, identifying the extreme values of a function within a specified interval is a fundamental task. This process, known as finding the absolute maximum and absolute minimum, has wide-ranging applications across various fields, including optimization problems, physics, engineering, and economics. In this article, we will delve into the function $f(x) = 3x^4 - 2x^3$ defined on the closed interval [-4, 4], with the goal of determining its absolute extrema. Understanding how to find these extreme values is crucial for anyone looking to optimize real-world scenarios or gain a deeper insight into the behavior of functions.

The process of finding the absolute maximum and absolute minimum values of a function involves several key steps. First, we must identify the critical points of the function within the given interval. Critical points are the locations where the function's derivative is either zero or undefined. These points are crucial because they represent potential turning points where the function's slope changes direction, indicating a local maximum or minimum. Second, we evaluate the function at these critical points, as well as at the endpoints of the interval. The endpoints are included because the absolute extrema can occur at the boundaries of the interval, even if there are no critical points there. Finally, by comparing the function values at all critical points and endpoints, we can determine the absolute maximum and absolute minimum values of the function over the interval.

This method, rooted in the Extreme Value Theorem, guarantees that a continuous function on a closed interval will attain both an absolute maximum and an absolute minimum value within that interval. The theorem provides a theoretical foundation for our approach, ensuring that we are not searching for values that do not exist. By systematically analyzing the function's derivative and evaluating it at relevant points, we can confidently identify the extreme values and gain a comprehensive understanding of the function's behavior within the specified domain. This is particularly important in applications where we need to find the optimal solution, whether it is the maximum profit, the minimum cost, or the most efficient design.

Finding Critical Points of $f(x) = 3x^4 - 2x^3$

To begin our exploration of the function $f(x) = 3x^4 - 2x^3$, our primary objective is to identify the critical points within the interval [-4, 4]. These critical points are the key to unlocking the function's extreme values, as they represent potential locations of local maxima or minima. The first step in this process involves calculating the derivative of the function, which will provide us with information about the function's slope and rate of change. The derivative, denoted as $f'(x)$, is found using the power rule of differentiation. Applying this rule to each term of the function, we get:

$f'(x) = 12x^3 - 6x^2$

This derivative function, $f'(x) = 12x^3 - 6x^2$, is a cubic polynomial that describes the slope of the original function $f(x)$ at any given point $x$. Critical points occur where the derivative is either equal to zero or is undefined. In this case, since the derivative is a polynomial, it is defined for all real numbers. Therefore, we only need to focus on finding the points where the derivative is equal to zero. To do this, we set $f'(x)$ equal to zero and solve for $x$:

$12x^3 - 6x^2 = 0$

To solve this equation, we can factor out the greatest common factor, which is $6x^2$:

$6x^2(2x - 1) = 0$

This factored form reveals the roots of the equation, which are the critical points of the function. Setting each factor equal to zero, we find:

$6x^2 = 0$ or $2x - 1 = 0$

Solving these equations, we obtain:

$x = 0$ or x = rac{1}{2}

Thus, we have identified two critical points for the function $f(x)$ within the interval [-4, 4]: $x = 0$ and x = rac{1}{2}. These points are crucial because they are potential locations of local maxima or minima. To determine whether these points are indeed local extrema, and to find the absolute extrema within the interval, we need to evaluate the function at these critical points, as well as at the endpoints of the interval.

Evaluating the Function at Critical Points and Endpoints

Having identified the critical points $x = 0$ and x = rac{1}{2}, the next crucial step in our analysis is to evaluate the function $f(x) = 3x^4 - 2x^3$ at these points, as well as at the endpoints of the interval [-4, 4]. This evaluation will provide us with the function values at these key locations, allowing us to compare them and ultimately determine the absolute maximum and absolute minimum values of the function within the specified interval. We begin by substituting each of these $x$-values into the function $f(x)$:

1. At $x = -4$ (left endpoint):

$f(-4) = 3(-4)^4 - 2(-4)^3 = 3(256) - 2(-64) = 768 + 128 = 896$

2. At $x = 0$ (critical point):

$f(0) = 3(0)^4 - 2(0)^3 = 0$

3. At x = rac{1}{2} (critical point):

f(rac{1}{2}) = 3(rac{1}{2})^4 - 2(rac{1}{2})^3 = 3(rac{1}{16}) - 2(rac{1}{8}) = rac{3}{16} - rac{2}{8} = rac{3}{16} - rac{4}{16} = -rac{1}{16}

4. At $x = 4$ (right endpoint):

$f(4) = 3(4)^4 - 2(4)^3 = 3(256) - 2(64) = 768 - 128 = 640$

Now that we have evaluated the function at all critical points and endpoints, we have the following values:

• $f(-4) = 896$
• $f(0) = 0$
• f(rac{1}{2}) = -rac{1}{16}
• $f(4) = 640$

By comparing these values, we can identify the absolute maximum and absolute minimum of the function within the interval [-4, 4]. The largest value corresponds to the absolute maximum, and the smallest value corresponds to the absolute minimum. This process of evaluation is crucial in optimization problems, as it allows us to pinpoint the exact locations where the function attains its extreme values.

Determining Absolute Maximum and Minimum Values

Having evaluated the function $f(x) = 3x^4 - 2x^3$ at the critical points and endpoints within the interval [-4, 4], we now possess the necessary information to determine the absolute maximum and absolute minimum values. The values we obtained are:

• $f(-4) = 896$
• $f(0) = 0$
• f(rac{1}{2}) = -rac{1}{16}
• $f(4) = 640$

To identify the absolute maximum, we simply look for the largest value among these. Comparing the values, we see that $f(-4) = 896$ is the largest. Therefore, the absolute maximum value of the function on the interval [-4, 4] is 896, and it occurs at $x = -4$. This means that the function reaches its highest point within the interval at $x = -4$, and that point has a y-coordinate of 896. The absolute maximum is a crucial concept in various applications, representing the highest possible value that a function can attain within a given domain.

Similarly, to find the absolute minimum, we look for the smallest value among the evaluated function values. Comparing the values, we observe that f(rac{1}{2}) = -rac{1}{16} is the smallest. Thus, the absolute minimum value of the function on the interval [-4, 4] is -rac{1}{16}, and it occurs at x = rac{1}{2}. This indicates that the function reaches its lowest point within the interval at x = rac{1}{2}, with a corresponding y-coordinate of -rac{1}{16}. The absolute minimum is equally important in many applications, representing the lowest possible value that a function can attain within a given domain.

In summary, by evaluating the function at the critical points and endpoints, we have successfully identified both the absolute maximum and absolute minimum values of $f(x) = 3x^4 - 2x^3$ on the interval [-4, 4]. The absolute maximum value is 896 at $x = -4$, and the absolute minimum value is -rac{1}{16} at x = rac{1}{2}. This process demonstrates the power of calculus in determining the extreme values of functions, which is a fundamental concept in optimization and other areas of mathematics and science.

Conclusion: Significance of Absolute Extrema

In conclusion, our comprehensive analysis of the function $f(x) = 3x^4 - 2x^3$ on the interval [-4, 4] has successfully revealed its extreme values. We found that the absolute maximum value is 896, occurring at $x = -4$, and the absolute minimum value is -rac{1}{16}, occurring at x = rac{1}{2}. This process involved several crucial steps, including finding the derivative of the function, identifying critical points, and evaluating the function at these critical points and the endpoints of the interval. By comparing the function values at these key locations, we were able to pinpoint the absolute extrema, which represent the highest and lowest points the function attains within the given interval.

The significance of finding absolute extrema extends far beyond the realm of pure mathematics. In various fields, such as engineering, economics, and physics, identifying extreme values is essential for optimization. For example, engineers might need to determine the maximum stress a structure can withstand, economists might seek to maximize profit or minimize cost, and physicists might be interested in finding the minimum energy state of a system. In all these scenarios, the ability to find the absolute maximum and absolute minimum of a function is a powerful tool.

The method we employed in this article, based on the principles of calculus and the Extreme Value Theorem, provides a systematic approach to finding absolute extrema. By understanding the behavior of a function's derivative, we can identify potential turning points and use them, along with the interval endpoints, to locate the extreme values. This approach is not only effective but also provides valuable insights into the function's overall behavior. The critical points, where the derivative is zero or undefined, represent locations where the function's slope changes direction, indicating potential local maxima or minima. The endpoints of the interval, on the other hand, represent the boundaries of the domain, where the function might attain its extreme values due to the constraints of the interval.

In summary, the ability to find absolute extrema is a fundamental skill in mathematics and a valuable tool in various applications. Our analysis of $f(x) = 3x^4 - 2x^3$ on the interval [-4, 4] has demonstrated the power of calculus in this context and highlighted the importance of understanding the behavior of functions and their derivatives. The absolute maximum and absolute minimum values provide crucial information about the function's range and its behavior within the specified domain, making this analysis essential for anyone working with mathematical models and optimization problems.