Absolute Extrema Of F(x) = 2sin(x) - √3x On [0, 2π]

Exploring the Realm of Trigonometric Functions: In the realm of mathematical analysis, the quest to determine the absolute extrema of a function stands as a fundamental endeavor. Absolute extrema, representing the highest and lowest points a function attains within a specified interval, provide critical insights into the function's behavior and characteristics. In this comprehensive exploration, we embark on a journey to unravel the absolute extrema of the function f(x) = 2sin(x) - √3x, defined within the interval [0, 2π]. This function, a harmonious blend of trigonometric and linear components, presents an intriguing challenge that necessitates the application of calculus principles and analytical techniques.

To embark on this quest, we first delve into the realm of critical points, those pivotal locations where the function's derivative gracefully vanishes or encounters discontinuity. These critical points serve as potential candidates for absolute extrema, acting as signposts along the function's trajectory. By meticulously calculating the derivative of f(x), we embark on a journey to pinpoint these critical points, laying the foundation for our subsequent analysis.

The derivative of f(x) = 2sin(x) - √3x, denoted as f'(x), is obtained through the application of differentiation rules. The derivative of 2sin(x) gracefully transforms into 2cos(x), while the derivative of -√3x elegantly simplifies to -√3. Thus, we arrive at the expression f'(x) = 2cos(x) - √3. The quest to unearth the critical points now hinges on solving the equation f'(x) = 0, which translates to 2cos(x) - √3 = 0. Rearranging this equation, we arrive at cos(x) = √3/2.

Pinpointing Critical Points within the Interval: The solutions to cos(x) = √3/2 within the interval [0, 2π] unveil the critical points that hold the key to our quest. Recall that the cosine function gracefully traces the x-coordinate of a point traversing the unit circle. The values of x for which cos(x) equals √3/2 correspond to the angles where the x-coordinate on the unit circle aligns with √3/2. These angles, elegantly revealed, are π/6 and 11π/6. Thus, we have successfully identified the critical points as x = π/6 and x = 11π/6. These points, along with the endpoints of the interval, x = 0 and x = 2π, form the constellation of potential absolute extrema.

The Grand Extrema Evaluation: With the critical points and endpoints in our grasp, we embark on the grand evaluation of f(x) at these pivotal locations. This evaluation will unveil the function's values at these critical junctures, allowing us to discern the absolute maximum and absolute minimum with unwavering certainty. The function's value at x = 0, denoted as f(0), gracefully simplifies to 2sin(0) - √3(0) = 0. At x = π/6, f(π/6) takes on the form 2sin(π/6) - √3(π/6) = 2(1/2) - √3(π/6) = 1 - √3π/6. At x = 11π/6, f(11π/6) elegantly transforms to 2sin(11π/6) - √3(11π/6) = 2(-1/2) - √3(11π/6) = -1 - 11√3π/6. Finally, at x = 2π, f(2π) gracefully simplifies to 2sin(2π) - √3(2π) = -2√3π.

Unveiling the Absolute Extrema: With the function's values meticulously evaluated at the critical points and endpoints, the moment of revelation arrives. Comparing these values, we discern the absolute maximum and absolute minimum with clarity. The absolute minimum, the lowest value attained by the function within the interval, elegantly emerges as f(11π/6) = -1 - 11√3π/6 ≈ -10.976. This minimum graces the point (11π/6, -10.976) on the function's graph. The absolute maximum, the highest value attained by the function, stands revealed as f(π/6) = 1 - √3π/6 ≈ 0.095. This maximum adorns the point (π/6, 0.095) on the function's graph.

Understanding Function Behavior: In delving deeper into the characteristics of the function f(x) = 2sin(x) - √3x within the interval [0, 2π], we embark on a comprehensive analysis that extends beyond merely identifying the absolute extrema. This analysis encompasses the function's increasing and decreasing intervals, its concavity, and the presence of any inflection points. By meticulously examining these aspects, we gain a profound understanding of the function's overall behavior and graphical representation.

To embark on this comprehensive analysis, we revisit the function's first derivative, f'(x) = 2cos(x) - √3. This derivative, as we previously established, plays a pivotal role in discerning the function's increasing and decreasing intervals. Recall that a positive derivative signifies an increasing function, while a negative derivative indicates a decreasing function. By meticulously analyzing the sign of f'(x) across the interval [0, 2π], we can precisely delineate the intervals where the function gracefully ascends and descends.

To determine the intervals where f'(x) > 0, we solve the inequality 2cos(x) - √3 > 0, which elegantly transforms into cos(x) > √3/2. Within the interval [0, 2π], this inequality holds true for x values nestled within the interval [0, π/6). Similarly, to determine the intervals where f'(x) < 0, we solve the inequality 2cos(x) - √3 < 0, which gracefully simplifies to cos(x) < √3/2. This inequality finds its domain in the intervals (π/6, 11π/6) and (11π/6, 2π]. Thus, we can confidently assert that f(x) gracefully increases on the interval [0, π/6) and gracefully decreases on the intervals (π/6, 11π/6) and (11π/6, 2π].

Unveiling Concavity and Inflection Points: To further enrich our understanding of the function's behavior, we turn our attention to the second derivative, denoted as f''(x). The second derivative unveils the function's concavity, revealing whether the function gracefully curves upwards or downwards. A positive second derivative signifies a concave-upward trajectory, while a negative second derivative indicates a concave-downward curvature. Points where the concavity gracefully transitions, known as inflection points, mark pivotal junctures in the function's graphical representation.

The second derivative of f(x), obtained by differentiating f'(x) = 2cos(x) - √3, elegantly emerges as f''(x) = -2sin(x). To determine the intervals where f''(x) > 0, we solve the inequality -2sin(x) > 0, which gracefully simplifies to sin(x) < 0. Within the interval [0, 2π], this inequality holds true for x values residing within the interval (π, 2π). Similarly, to determine the intervals where f''(x) < 0, we solve the inequality -2sin(x) < 0, which elegantly transforms into sin(x) > 0. This inequality finds its domain in the interval (0, π). Thus, we can confidently assert that f(x) gracefully curves upwards on the interval (π, 2π) and gracefully curves downwards on the interval (0, π).

Inflection Points Unveiled: Inflection points, those pivotal locations where the concavity gracefully transitions, occur when f''(x) = 0 or when f''(x) is undefined. In this context, we seek solutions to the equation -2sin(x) = 0, which elegantly simplifies to sin(x) = 0. Within the interval [0, 2π], the solutions to this equation gracefully emerge as x = 0, x = π, and x = 2π. These points, where the function's concavity undergoes a graceful transformation, mark the inflection points on the function's graph.

Evaluating the function at these inflection points, we find f(0) = 0, f(π) = -√3π, and f(2π) = -2√3π. Thus, the inflection points gracefully adorn the coordinates (0, 0), (π, -√3π), and (2π, -2√3π). These points, where the function's curvature gracefully shifts, provide valuable insights into the function's overall graphical representation.

Embarking on the Extrema Quest: In the realm of mathematical analysis, the determination of absolute extrema stands as a fundamental pursuit. Absolute extrema, representing the highest and lowest points a function attains within a specified interval, provide crucial insights into the function's behavior and characteristics. To embark on this quest, we present a step-by-step guide that elegantly unveils the process of determining absolute extrema.

Step 1: The Derivative Revelation: The first step in our quest involves the elegant calculation of the function's derivative. The derivative, denoted as f'(x), holds the key to unraveling the function's critical points, those pivotal locations where the function's slope gracefully vanishes or encounters discontinuity. These critical points serve as potential candidates for absolute extrema, acting as signposts along the function's trajectory.

Step 2: Critical Point Unveiling: With the derivative in our grasp, the next step involves the meticulous identification of critical points. Critical points gracefully emerge as the solutions to the equation f'(x) = 0, or as points where f'(x) gracefully encounters discontinuity. These points, where the function's slope undergoes a dramatic transformation, hold the promise of harboring absolute extrema.

Step 3: Endpoint Embrace: In addition to the critical points, the endpoints of the interval under consideration also claim their place as potential absolute extrema. These endpoints, representing the boundaries of the function's domain, may very well harbor the function's highest or lowest values.

Step 4: The Grand Evaluation: With the critical points and endpoints in our possession, we embark on the grand evaluation of the function at these pivotal locations. This evaluation will unveil the function's values at these critical junctures, allowing us to discern the absolute maximum and absolute minimum with unwavering certainty.

Step 5: Extrema Revelation: With the function's values meticulously evaluated at the critical points and endpoints, the moment of revelation arrives. Comparing these values, we discern the absolute maximum, the highest value attained by the function within the interval, and the absolute minimum, the lowest value attained by the function within the interval.

In this comprehensive exploration, we have successfully unraveled the absolute extrema of the function f(x) = 2sin(x) - √3x within the interval [0, 2π]. Through the meticulous application of calculus principles and analytical techniques, we have identified the absolute minimum as approximately -10.976, gracing the point (11π/6, -10.976), and the absolute maximum as approximately 0.095, adorning the point (π/6, 0.095). These extrema, representing the function's lowest and highest points within the specified interval, provide invaluable insights into the function's behavior and characteristics.

Furthermore, our comprehensive analysis extended beyond merely identifying the absolute extrema. We delved into the function's increasing and decreasing intervals, its concavity, and the presence of inflection points. This holistic examination painted a vivid picture of the function's graphical representation and overall behavior. The function gracefully increases on the interval [0, π/6), decreases on the intervals (π/6, 11π/6) and (11π/6, 2π], curves downwards on the interval (0, π), curves upwards on the interval (π, 2π), and exhibits inflection points at (0, 0), (π, -√3π), and (2π, -2√3π). This comprehensive analysis serves as a testament to the power of calculus in unraveling the intricacies of function behavior.

The step-by-step guide presented in this exploration serves as a valuable tool for determining absolute extrema across a wide range of functions. By meticulously following these steps, one can confidently embark on the quest to unearth the highest and lowest points a function attains within a specified interval. The determination of absolute extrema stands as a fundamental concept in mathematical analysis, with applications spanning diverse fields, including optimization problems, physics, and engineering. Understanding and mastering the techniques for finding absolute extrema empowers individuals to solve real-world problems and gain a deeper appreciation for the elegance and power of mathematics.