Finding Absolute Extrema Of F(x) = 5(2x)^x On [0.1, 1]

Introduction to Finding Absolute Extreme Values

In the realm of calculus, a crucial task is to determine the absolute extreme values of a function within a given interval. These values, representing the maximum and minimum outputs of the function, are vital in various applications, ranging from optimization problems in engineering to economic modeling. This article delves into the process of finding these extreme values, focusing on the function f(x) = 5(2x)^x within the interval [0.1, 1]. We will explore the theoretical underpinnings, the steps involved, and the practical implications of identifying these critical points. Understanding how to locate and evaluate absolute extrema is fundamental for anyone seeking a deeper grasp of calculus and its applications. The absolute extreme values provide a comprehensive understanding of the function's behavior, allowing us to identify its highest and lowest points within the specified domain. This knowledge is indispensable for solving optimization problems and gaining insights into the function's overall characteristics. By meticulously analyzing the function and its derivative, we can pinpoint the precise locations of these extrema and determine their corresponding values. This process not only enhances our understanding of the function itself but also strengthens our problem-solving skills in calculus and related fields. The concepts discussed in this article form the bedrock of many advanced mathematical and scientific applications, making it an essential topic for students and professionals alike.

Understanding the Function f(x) = 5(2x)^x

Before diving into the calculus, let's first understand the function itself: f(x) = 5(2x)^x. This is a transcendental function, meaning it involves a variable in the exponent. This type of function exhibits unique behavior and requires careful analysis. The domain of this function is restricted by the term (2x)^x, which is defined for x > 0. Our interval of interest, [0.1, 1], falls within this domain, so the function is well-defined within the specified range. Understanding the nature of this function is crucial for determining the appropriate methods for finding its extreme values. Transcendental functions often lack straightforward algebraic solutions, necessitating the use of calculus techniques such as differentiation to analyze their behavior. The presence of the variable in the exponent introduces a level of complexity that requires a thorough understanding of logarithmic differentiation and its applications. By grasping the fundamental properties of exponential and logarithmic functions, we can effectively tackle the challenges posed by this type of function and accurately identify its critical points. Furthermore, visualizing the graph of the function can provide valuable insights into its behavior and potential extrema, aiding in the analytical process.

Calculus Techniques for Finding Extrema

To find the absolute extreme values, we employ the following calculus techniques:

1. Find the Derivative: The first step is to find the derivative of f(x), denoted as f'(x). This derivative will help us identify critical points where the function's slope is zero or undefined.
2. Identify Critical Points: Critical points are the values of x where f'(x) = 0 or f'(x) is undefined. These points are potential locations of local maxima or minima.
3. Evaluate at Endpoints and Critical Points: The absolute extreme values can occur either at the critical points within the interval or at the endpoints of the interval. Therefore, we need to evaluate f(x) at all critical points and the endpoints of [0.1, 1].
4. Determine Absolute Extrema: By comparing the function values obtained in the previous step, we can identify the absolute maximum and absolute minimum values within the interval.

These techniques are rooted in the fundamental theorems of calculus, which provide the theoretical framework for analyzing the behavior of functions. The derivative plays a crucial role in identifying the function's increasing and decreasing intervals, as well as the points where it changes direction. Critical points, where the derivative is zero or undefined, are potential locations of local extrema, but it is essential to evaluate the function at these points and the endpoints of the interval to determine the absolute extrema. The Extreme Value Theorem guarantees the existence of absolute extrema for continuous functions on closed intervals, making this process both rigorous and effective. By systematically applying these calculus techniques, we can confidently determine the absolute extreme values of a function within a given domain.

Step-by-Step Solution for f(x) = 5(2x)^x

Let's apply these techniques to our function f(x) = 5(2x)^x on the interval [0.1, 1].

1. Finding the Derivative

To find the derivative f'(x), we use logarithmic differentiation. First, take the natural logarithm of both sides:

ln(f(x)) = ln(5(2x)^x)

Using logarithm properties, we get:

ln(f(x)) = ln(5) + ln((2x)^x)

ln(f(x)) = ln(5) + xln(2x)

Now, differentiate both sides with respect to x:

(1/f(x)) f'(x) = 0 + ln(2x) + x(1/(2x))*2

(1/f(x)) f'(x) = ln(2x) + 1

Multiply both sides by f(x) to isolate f'(x):

f'(x) = f(x) (ln(2x) + 1)

Substitute f(x) = 5(2x)^x:

f'(x) = 5(2x)^x (ln(2x) + 1)

The process of finding the derivative using logarithmic differentiation is crucial for functions where the variable appears in the exponent. This technique simplifies the differentiation process by converting the exponential function into a logarithmic one, which can be handled more easily using standard differentiation rules. By carefully applying the chain rule and the properties of logarithms, we can accurately determine the derivative of complex functions like f(x) = 5(2x)^x. The resulting derivative, f'(x), provides valuable information about the function's slope and behavior, allowing us to identify critical points and ultimately determine the absolute extrema. A thorough understanding of logarithmic differentiation is essential for tackling a wide range of calculus problems involving exponential and logarithmic functions.

2. Identifying Critical Points

Critical points occur where f'(x) = 0 or f'(x) is undefined. Since 5(2x)^x is always positive on [0.1, 1], we only need to consider where (ln(2x) + 1) = 0:

ln(2x) + 1 = 0

ln(2x) = -1

2x = e^-1

x = e^-1 / 2

x ≈ 0.1839

The critical point x ≈ 0.1839 lies within our interval [0.1, 1]. The process of identifying critical points is a cornerstone of finding extreme values. These points represent potential locations of local maxima, minima, or saddle points, where the function's slope changes direction. By setting the derivative equal to zero and solving for x, we can pinpoint these crucial points and further analyze their significance. In this case, we found a single critical point within the interval, but it's essential to consider all possible critical points, including those where the derivative is undefined. The critical points, along with the endpoints of the interval, form the set of candidate points for the absolute extrema. By evaluating the function at these points, we can determine the absolute maximum and minimum values within the specified domain. The careful identification and analysis of critical points are fundamental for solving optimization problems and gaining a comprehensive understanding of a function's behavior.

3. Evaluating at Endpoints and Critical Points

Now, we evaluate f(x) at the endpoints of the interval [0.1, 1] and the critical point x ≈ 0.1839:

• f(0.1) = 5(2 * 0.1)^0.1 ≈ 5(0.2)^0.1 ≈ 4.269
• f(1) = 5(2 * 1)^1 = 5(2) = 10
• f(0.1839) = 5(2 * 0.1839)^0.1839 ≈ 5(0.3678)^0.1839 ≈ 3.944

The evaluation of the function at the endpoints and critical points is a crucial step in determining the absolute extreme values. This process involves substituting the x-values of these points into the original function and calculating the corresponding y-values. By comparing these function values, we can identify the absolute maximum and minimum within the given interval. The endpoints represent the boundaries of the domain, while the critical points represent potential local extrema within the domain. It's essential to evaluate the function at all these points to ensure that we capture the absolute extreme values, which may occur at any of these locations. The accuracy of these evaluations is paramount, as even small errors can lead to incorrect conclusions about the function's behavior. The process of evaluation provides a concrete understanding of the function's range within the specified interval and allows us to definitively determine its highest and lowest points.

4. Determining Absolute Extrema

Comparing the values, we see that:

• The absolute minimum value is approximately 3.944 at x ≈ 0.1839.
• The absolute maximum value is 10 at x = 1.

The final step in finding the absolute extreme values is to compare the function values obtained at the endpoints and critical points. By examining these values, we can definitively identify the absolute maximum and absolute minimum within the given interval. The absolute maximum represents the highest point the function reaches within the domain, while the absolute minimum represents the lowest point. In this case, we found that the absolute minimum occurs at the critical point x ≈ 0.1839, while the absolute maximum occurs at the endpoint x = 1. This comparison highlights the importance of evaluating the function at both critical points and endpoints to ensure that we capture the true extreme values. The determination of absolute extrema is a fundamental concept in calculus and has wide-ranging applications in optimization problems and other areas of mathematics and science. By systematically following the steps outlined in this article, we can confidently find the absolute extreme values of a function within a specified interval.

Conclusion

Therefore, for the function f(x) = 5(2x)^x on the interval [0.1, 1], the absolute minimum value is approximately 3.944 at x ≈ 0.1839, and the absolute maximum value is 10 at x = 1. This methodical approach, involving differentiation, critical point identification, and evaluation, provides a robust method for finding absolute extrema of functions within specified intervals. The process of determining absolute extreme values is a cornerstone of calculus and has far-reaching applications in various fields. By understanding the theoretical underpinnings and systematically applying the techniques discussed in this article, we can confidently tackle optimization problems and gain valuable insights into the behavior of functions. The ability to find absolute extrema is essential for solving real-world problems in engineering, economics, and other disciplines, making it a fundamental skill for students and professionals alike. The insights gained from this analysis can inform decision-making processes and lead to optimal solutions in a wide range of contexts.