Find Intervals Of Increasing And Decreasing F(x) = -2cos(x) - √2x On [0, Π]

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Understanding the behavior of functions, specifically identifying intervals where they increase or decrease, is a fundamental concept in calculus. This article delves into the process of determining these intervals for the function f(x) = -2cos(x) - √2x within the closed interval [0, π]. By employing the first derivative test, we will meticulously analyze the function's slope to pinpoint its increasing and decreasing nature. This exploration will not only enhance your comprehension of calculus principles but also equip you with the practical skills to analyze a wide range of functions.

1. Introduction to Increasing and Decreasing Functions

At its core, the concept of increasing and decreasing functions describes how the output of a function changes as its input varies. A function is said to be increasing on an interval if its output values rise as the input values increase. Conversely, a function is decreasing if its output values fall as the input values increase. Visually, an increasing function slopes upwards from left to right, while a decreasing function slopes downwards. These behaviors are not only crucial for understanding a function's graph but also for various applications in optimization problems and modeling real-world phenomena.

To formally define these concepts, let's consider a function f defined on an interval I. We say that f is increasing on I if for any two numbers x₁ and x₂ in I, where x₁ < x₂, we have f(x₁) < f(x₂). Similarly, f is decreasing on I if for any two numbers x₁ and x₂ in I, where x₁ < x₂, we have f(x₁) > f(x₂). These inequalities precisely capture the intuitive notion of a function's output rising or falling with increasing input.

The key to identifying these intervals lies in the function's derivative. The derivative, denoted as f'(x), provides the instantaneous rate of change of the function at a given point. A positive derivative indicates that the function is increasing at that point, while a negative derivative indicates that the function is decreasing. A zero derivative suggests a stationary point, which could be a local maximum, local minimum, or a point of inflection. This relationship between the derivative and the function's behavior forms the basis of the first derivative test, a powerful tool for analyzing functions.

2. The First Derivative Test: A Powerful Tool

The first derivative test is a cornerstone technique in calculus for determining the intervals where a function is increasing or decreasing. It leverages the relationship between a function's derivative and its slope. Essentially, the sign of the derivative tells us whether the function is rising (positive derivative), falling (negative derivative), or momentarily flat (zero derivative). This test is not only valuable for understanding the function's behavior but also for locating local extrema, such as maxima and minima.

The first step in applying the first derivative test is to find the derivative of the function, f'(x). This derivative represents the instantaneous rate of change of the function at any point x. Once we have the derivative, we need to identify the critical points of the function. These are the points where the derivative is either equal to zero or undefined. Critical points are crucial because they mark potential transitions between increasing and decreasing intervals.

Next, we construct a sign chart for the derivative. This involves creating a number line and marking the critical points on it. These critical points divide the number line into intervals. Within each interval, we choose a test value and evaluate the derivative at that point. The sign of the derivative at the test value tells us whether the function is increasing or decreasing throughout that entire interval. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing; and if it's zero, we have a stationary point. By analyzing the sign chart, we can definitively determine the intervals where the function is increasing and decreasing.

Furthermore, the first derivative test can help us classify critical points as local maxima or minima. If the derivative changes from positive to negative at a critical point, it indicates a local maximum (a peak in the graph). Conversely, if the derivative changes from negative to positive, it signifies a local minimum (a valley in the graph). These insights into the function's local behavior add another layer of understanding to its overall characteristics.

3. Applying the First Derivative Test to f(x) = -2cos(x) - √2x

Let's now apply the first derivative test to our specific function, f(x) = -2cos(x) - √2x, defined on the interval [0, π]. Our goal is to identify the intervals where this function is increasing and decreasing. This process involves finding the derivative, determining critical points, creating a sign chart, and interpreting the results.

First, we need to find the derivative of f(x). Using the rules of differentiation, we have:

f'(x) = d/dx (-2cos(x) - √2x) = 2sin(x) - √2

This derivative, f'(x) = 2sin(x) - √2, represents the slope of the function at any point x within the interval [0, π].

Next, we need to find the critical points, which are the points where f'(x) = 0 or f'(x) is undefined. In this case, f'(x) is a continuous function, so it is never undefined. We need to solve the equation 2sin(x) - √2 = 0:

2sin(x) = √2 sin(x) = √2 / 2

The solutions to this equation within the interval [0, π] are x = π/4 and x = 3π/4. These are our critical points. They divide the interval [0, π] into three subintervals: [0, π/4), (π/4, 3π/4), and (3π/4, π].

Now, we construct a sign chart for f'(x). We choose a test value within each interval and evaluate f'(x) at that point:

  • Interval [0, π/4): Let's choose x = 0. f'(0) = 2sin(0) - √2 = -√2, which is negative.
  • Interval (π/4, 3π/4): Let's choose x = π/2. f'(π/2) = 2sin(π/2) - √2 = 2 - √2, which is positive.
  • Interval (3π/4, π]: Let's choose x = π. f'(π) = 2sin(π) - √2 = -√2, which is negative.

Based on the sign chart, we can conclude that:

  • f(x) is decreasing on the interval [0, π/4) because f'(x) is negative.
  • f(x) is increasing on the interval (π/4, 3π/4) because f'(x) is positive.
  • f(x) is decreasing on the interval (3π/4, π] because f'(x) is negative.

4. Analyzing the Intervals of Increase and Decrease

Having applied the first derivative test, we've successfully identified the intervals where our function, f(x) = -2cos(x) - √2x, is increasing and decreasing on the interval [0, π]. This analysis provides valuable insights into the function's behavior and its graphical representation. Understanding these intervals allows us to sketch a more accurate graph and predict the function's values within the given domain.

As we found, the function is decreasing on the interval [0, π/4). This means that as x increases from 0 to π/4, the value of f(x) decreases. The negative derivative in this interval confirms that the function's slope is downward. This decreasing behavior continues until we reach the critical point x = π/4.

At x = π/4, the derivative changes sign, and the function transitions to an increasing phase on the interval (π/4, 3π/4). Here, as x increases from π/4 to 3π/4, the value of f(x) also increases. The positive derivative indicates an upward slope. This increasing behavior persists until we reach the next critical point at x = 3π/4.

Beyond x = 3π/4, the function once again decreases on the interval (3π/4, π]. As x increases from 3π/4 to π, the value of f(x) decreases, and the derivative becomes negative again, indicating a downward slope. This completes the analysis of the function's increasing and decreasing behavior across the given interval.

Furthermore, the critical points x = π/4 and x = 3π/4 provide information about local extrema. At x = π/4, the function changes from decreasing to increasing, suggesting a local minimum. Conversely, at x = 3π/4, the function changes from increasing to decreasing, indicating a local maximum. These local extrema are important features of the function's graph and can be useful in various optimization applications.

5. Visualizing the Function's Behavior

To solidify our understanding, visualizing the function's graph is immensely helpful. Sketching the graph of f(x) = -2cos(x) - √2x on the interval [0, π] allows us to see the increasing and decreasing intervals in action. The graph will visually confirm our analytical findings and provide a more intuitive grasp of the function's behavior.

Based on our analysis, we expect the graph to start with a downward slope from x = 0 until it reaches a minimum point at x = π/4. From there, the graph should slope upwards, reaching a maximum point at x = 3π/4. Finally, the graph should slope downwards again until it reaches x = π.

The minimum point at x = π/4 corresponds to a local minimum value of f(π/4) = -2cos(π/4) - √2(π/4) = -√2 - √2(π/4). Similarly, the maximum point at x = 3π/4 corresponds to a local maximum value of f(3π/4) = -2cos(3π/4) - √2(3π/4) = √2 - √2(3π/4). These values provide specific coordinates for key points on the graph.

By plotting these points and considering the increasing and decreasing intervals, we can create a rough sketch of the graph. The sketch will show a curve that initially decreases, reaches a minimum, increases to a maximum, and then decreases again. This visual representation reinforces the conclusions we drew from the first derivative test and enhances our understanding of the function's behavior.

Moreover, graphing tools can be used to generate a precise plot of the function, allowing for a more detailed examination of its characteristics. Such tools can confirm our analysis and reveal any subtle nuances in the function's behavior that might not be immediately apparent from the sketch. The combination of analytical techniques and visual representations provides a comprehensive understanding of the function.

6. Conclusion: Mastering Increasing and Decreasing Intervals

In conclusion, determining the intervals where a function increases or decreases is a fundamental skill in calculus, with wide-ranging applications in mathematics, science, and engineering. By mastering the first derivative test, we gain a powerful tool for analyzing the behavior of functions and understanding their graphical representations. This article has demonstrated the step-by-step process of applying this test to the function f(x) = -2cos(x) - √2x on the interval [0, π], revealing its intervals of increase and decrease.

We began by defining the concepts of increasing and decreasing functions, emphasizing the role of the derivative in determining a function's slope. We then delved into the first derivative test, outlining the steps involved in finding the derivative, identifying critical points, constructing a sign chart, and interpreting the results. This test provides a systematic approach to analyzing a function's behavior.

Applying the first derivative test to our specific function, we found that f(x) is decreasing on the interval [0, π/4), increasing on the interval (π/4, 3π/4), and decreasing again on the interval (3π/4, π]. These intervals provide a clear picture of how the function's output changes as its input varies. Furthermore, the critical points x = π/4 and x = 3π/4 were identified as locations of a local minimum and a local maximum, respectively.

Visualizing the function's graph reinforces our analytical findings and provides a more intuitive understanding of its behavior. The graph confirms the increasing and decreasing intervals and highlights the local extrema. By combining analytical techniques with visual representations, we achieve a comprehensive understanding of the function.

The ability to analyze functions in this way is essential for solving optimization problems, modeling real-world phenomena, and gaining a deeper appreciation for the power of calculus. By understanding the principles and techniques discussed in this article, you are well-equipped to tackle a wide range of function analysis challenges.