Proving F(x) = (16 Sin X) / (4 + Cos X) - X Is Strictly Decreasing On (π/2, Π)

by Admin 79 views

Introduction

In the realm of mathematical analysis, understanding the behavior of functions is crucial. One key aspect of this understanding is determining whether a function is strictly increasing, strictly decreasing, or neither, within a given interval. This article delves into the fascinating world of calculus to demonstrate that the function f(x) = (16 sin x) / (4 + cos x) - x exhibits a strictly decreasing nature within the interval (π/2, π). This exploration will involve employing the principles of differential calculus, specifically the concept of the derivative, to analyze the function's rate of change. By examining the sign of the derivative within the specified interval, we can definitively establish the function's monotonic behavior. This analysis not only enhances our comprehension of this particular function but also reinforces the broader applicability of calculus in dissecting the properties of mathematical expressions.

Understanding Strictly Decreasing Functions

A strictly decreasing function is a function where, as the input x increases, the output f(x) decreases. More formally, a function f(x) is strictly decreasing on an interval (a, b) if for any two points x₁ and x₂ in (a, b), where x₁ < x₂, we have f(x₁) > f(x₂). Graphically, this means that as you move from left to right along the graph of the function, the curve slopes downwards. This concept is fundamental in various areas of mathematics, including optimization problems, where identifying intervals of decrease helps pinpoint minimum values. Understanding the behavior of functions, such as whether they are increasing or decreasing, is crucial in modeling real-world phenomena, from population growth to the decay of radioactive substances. In essence, the notion of strictly decreasing functions provides a powerful tool for analyzing and predicting trends within mathematical models and their applications.

The Role of the Derivative in Determining Monotonicity

The derivative of a function, denoted as f'(x), plays a pivotal role in determining the function's monotonicity – whether it is increasing, decreasing, or constant. The derivative represents the instantaneous rate of change of the function at a given point. If f'(x) > 0 on an interval, the function is strictly increasing on that interval, indicating that the function's value is growing as x increases. Conversely, if f'(x) < 0 on an interval, the function is strictly decreasing on that interval, signifying that the function's value is diminishing as x increases. When f'(x) = 0 on an interval, the function is constant on that interval, meaning its value remains unchanged as x varies. This relationship between the derivative and the function's behavior forms the cornerstone of many calculus applications, including finding local maxima and minima, analyzing the concavity of curves, and optimizing functions in various contexts. By examining the sign of the derivative, we gain valuable insights into the function's overall trend and its behavior within specific intervals.

Calculating the Derivative of f(x)

To demonstrate that the function f(x) = (16 sin x) / (4 + cos x) - x is strictly decreasing in the interval (π/2, π), our first crucial step is to calculate its derivative, f'(x). This process involves applying the rules of differential calculus, specifically the quotient rule and the chain rule. The quotient rule is essential for differentiating the term (16 sin x) / (4 + cos x), as it involves a ratio of two functions. The chain rule may also be necessary depending on the complexity of the functions involved. Let's break down the process:

  1. Identify the components: We have a function f(x) that is the difference of two terms: a quotient and a simple x term. The derivative of x is straightforward, which is 1. The main challenge lies in differentiating the quotient.
  2. Apply the quotient rule: The quotient rule states that if we have a function u(x) / v(x), its derivative is (v(x)u'(x) - u(x)v'(x)) / (v(x))². In our case, u(x) = 16 sin x and v(x) = 4 + cos x.
  3. Find the derivatives of u(x) and v(x):
    • u'(x) = 16 cos x
    • v'(x) = -sin x
  4. Apply the quotient rule formula:
    • The derivative of (16 sin x) / (4 + cos x) is (((4 + cos x)(16 cos x) - (16 sin x)(-sin x)) / (4 + cos x)²)
  5. Simplify the expression:
    • Expand the numerator: (64 cos x + 16 cos²x + 16 sin²x) / (4 + cos x)²
    • Use the trigonometric identity sin²x + cos²x = 1: (64 cos x + 16(cos²x + sin²x)) / (4 + cos x)² = (64 cos x + 16) / (4 + cos x)²
  6. Differentiate the x term: The derivative of -x is simply -1.
  7. Combine the results:
    • f'(x) = (64 cos x + 16) / (4 + cos x)² - 1

By meticulously applying the quotient rule and simplifying the resulting expression, we have successfully determined the derivative of f(x). This derivative, f'(x) = (64 cos x + 16) / (4 + cos x)² - 1, is now our key tool for analyzing the function's monotonicity within the specified interval.

Analyzing the Sign of f'(x) in (π/2, π)

Now that we have the derivative, f'(x) = (64 cos x + 16) / (4 + cos x)² - 1, the next critical step is to analyze its sign within the interval (π/2, π). This analysis will definitively reveal whether the function f(x) is strictly decreasing in this interval. To achieve this, we need to determine if f'(x) is negative for all x values within (π/2, π). This involves examining the behavior of the cosine function within this interval and how it influences the overall sign of f'(x). Let's break down the process:

  1. Understand the behavior of cos x in (π/2, π): Within the interval (π/2, π), the cosine function, cos x, takes on negative values. Specifically, it ranges from 0 at π/2 to -1 at π. This characteristic of cos x is crucial in determining the sign of f'(x).
  2. Analyze the numerator of the first term: The numerator of the first term in f'(x) is (64 cos x + 16). Since cos x is negative in (π/2, π), 64 cos x will also be negative. The magnitude of 64 cos x will depend on the value of cos x. As cos x ranges from 0 to -1, 64 cos x will range from 0 to -64. Therefore, (64 cos x + 16) will be negative when 64 cos x < -16, which means cos x < -1/4.
  3. Analyze the denominator of the first term: The denominator of the first term, (4 + cos x)², is always positive because it is a square. Even though cos x is negative in (π/2, π), adding 4 to it and then squaring the result ensures positivity. This positive denominator plays a vital role in determining the sign of the overall first term.
  4. Consider the entire f'(x) expression: f'(x) = (64 cos x + 16) / (4 + cos x)² - 1. We need to show that this entire expression is negative in the interval (π/2, π). To do this, we can analyze when (64 cos x + 16) / (4 + cos x)² < 1.
  5. Manipulate the inequality:
    • (64 cos x + 16) / (4 + cos x)² < 1
    • 64 cos x + 16 < (4 + cos x)²
    • 64 cos x + 16 < 16 + 8 cos x + cos²x
    • 0 < cos²x - 56 cos x
    • 0 < cos x (cos x - 56)
  6. Interpret the inequality: Since cos x is negative in (π/2, π), the term (cos x - 56) will always be negative (as cos x is between -1 and 0). Therefore, the product cos x (cos x - 56) will always be positive in (π/2, π). This confirms that f'(x) < 0 in this interval.

Through this detailed analysis, we have conclusively shown that f'(x) is negative within the interval (π/2, π). This crucial finding directly leads to the conclusion that the function f(x) is strictly decreasing in this interval.

Conclusion

In conclusion, we have successfully demonstrated that the function f(x) = (16 sin x) / (4 + cos x) - x is strictly decreasing in the interval (π/2, π). This was achieved by employing the principles of differential calculus. We first calculated the derivative of the function, f'(x), using the quotient rule and basic differentiation rules. Then, we meticulously analyzed the sign of f'(x) within the specified interval. By understanding the behavior of the cosine function in (π/2, π) and carefully manipulating inequalities, we showed that f'(x) < 0 for all x in (π/2, π). This negative derivative unequivocally establishes that f(x) is strictly decreasing in this interval. This exercise not only provides a concrete example of how calculus can be used to analyze the behavior of functions but also reinforces the importance of derivatives in determining monotonicity, a fundamental concept in mathematical analysis. Understanding the decreasing nature of this function has implications in various applications, including optimization problems and modeling real-world phenomena where decreasing trends are observed.