Find Intervals Of Increase And Decrease For F(x) = Sin(x) + Cos(x)

In calculus, determining the intervals where a function is increasing or decreasing is a fundamental concept. This involves analyzing the first derivative of the function. By identifying where the derivative is positive (increasing) or negative (decreasing), we can understand the function's behavior over a given interval. This article provides a detailed walkthrough on how to find the open intervals where the function $f(x) = \sin(x) + \cos(x)$ is increasing or decreasing within the interval $0 < x < 2\pi$. This analysis is crucial for understanding the function's graph, identifying local maxima and minima, and grasping the overall behavior of the function.

Understanding Increasing and Decreasing Functions

Before diving into the specifics of our function, it's important to grasp the core concepts of increasing and decreasing functions. A function is said to be increasing on an interval if its values rise as the input (x-value) increases. Conversely, a function is decreasing on an interval if its values fall as the input increases. Graphically, an increasing function slopes upwards from left to right, while a decreasing function slopes downwards. The key to identifying these intervals lies in the function's first derivative. The first derivative, denoted as $f'(x)$, gives us the instantaneous rate of change of the function at any point. A positive $f'(x)$ indicates that the function is increasing, a negative $f'(x)$ indicates that the function is decreasing, and $f'(x) = 0$ indicates a critical point where the function may have a local maximum, local minimum, or a saddle point. These critical points are where the function's direction changes, making them essential for interval analysis. Understanding these basic principles allows us to methodically approach the problem of finding where a function increases or decreases.

Step-by-Step Approach to Finding Intervals

To methodically identify the intervals where a function is increasing or decreasing, a step-by-step approach is crucial. This process ensures accuracy and clarity in our analysis. We'll break down the method into five key steps, each building on the previous one to provide a comprehensive understanding of the function's behavior. First, we need to calculate the first derivative of the function. This derivative, denoted as $f'(x)$, represents the instantaneous rate of change of the function and is the foundation for determining increasing and decreasing intervals. Second, we find the critical points of the function. These are the points where the first derivative is either equal to zero or undefined. Critical points are crucial because they mark potential turning points where the function changes from increasing to decreasing or vice versa. Third, we create a number line and mark the critical points on it. This number line helps visualize the intervals created by the critical points and provides a framework for testing the behavior of the derivative in each interval. Fourth, we choose test values within each interval and evaluate the first derivative at these points. The sign of the derivative (positive or negative) in each interval tells us whether the function is increasing or decreasing in that interval. Finally, based on the sign analysis of the derivative, we conclude the intervals where the function is increasing and decreasing. This structured approach ensures that we consider all critical points and intervals, leading to a complete understanding of the function's behavior.

Applying the Method to f(x) = sin(x) + cos(x)

Now, let's apply the step-by-step method to the function $f(x) = \sin(x) + \cos(x)$ within the interval $0 < x < 2\pi$. This example will provide a practical demonstration of how to find increasing and decreasing intervals for a trigonometric function. This involves differentiating the function, finding critical points, constructing a number line, testing intervals, and finally, stating the intervals where the function is increasing or decreasing.

1. Calculate the First Derivative

The first step in our analysis is to find the first derivative of the function $f(x) = \sin(x) + \cos(x)$. Recall that the derivative of $\sin(x)$ is $\cos(x)$ and the derivative of $\cos(x)$ is $-\sin(x)$. Applying these rules, we find the derivative of $f(x)$:

$f'(x) = \frac{d}{dx}(\sin(x) + \cos(x)) = \frac{d}{dx}(\sin(x)) + \frac{d}{dx}(\cos(x)) = \cos(x) - \sin(x)$

Thus, the first derivative of our function is $f'(x) = \cos(x) - \sin(x)$. This derivative is essential because its sign will tell us where the original function $f(x)$ is increasing or decreasing. A positive $f'(x)$ indicates that $f(x)$ is increasing, while a negative $f'(x)$ indicates that $f(x)$ is decreasing. This calculation is the crucial first step in understanding the behavior of our function.

2. Find the Critical Points

The next crucial step is to find the critical points of the function. Critical points are the values of $x$ where the first derivative, $f'(x)$, is either equal to zero or undefined. These points are significant because they often mark the transitions between increasing and decreasing intervals. To find the critical points, we set $f'(x) = 0$ and solve for $x$ within the given interval $0 < x < 2\pi$:

$\cos(x) - \sin(x) = 0$

To solve this equation, we can rearrange it as follows:

$\cos(x) = \sin(x)$

Now, divide both sides by $\cos(x)$, provided $\cos(x) \neq 0$:

$1 = \frac{\sin(x)}{\cos(x)}$

Recall that $\frac{\sin(x)}{\cos(x)} = \tan(x)$, so we have:

$\tan(x) = 1$

Within the interval $0 < x < 2\pi$, the tangent function equals 1 at two points: $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$. These are the solutions to our equation and therefore the critical points of the function within the specified interval. It's important to note that we considered the condition $\cos(x) \neq 0$. If $\cos(x)$ were zero, then $\tan(x)$ would be undefined, which could also indicate critical points. However, in this case, the points where $\cos(x) = 0$ (i.e., $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$) do not satisfy the original equation $\cos(x) = \sin(x)$, so they are not critical points for our function. Thus, our critical points are $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$, and they play a key role in determining the intervals of increase and decrease.

3. Create a Number Line

Now that we've identified the critical points, the next step is to create a number line. This visual tool helps us organize the intervals created by the critical points and test the behavior of the first derivative, $f'(x)$, within each interval. Draw a horizontal line and mark the critical points, $x = \frac{\pi}{4}$ and $x = \frac{5\pi}{4}$, on the line. These points divide the interval $0 < x < 2\pi$ into three subintervals:

1. $(0, \frac{\pi}{4})$
2. $(\frac{\pi}{4}, \frac{5\pi}{4})$
3. $(\frac{5\pi}{4}, 2\pi)$

The number line provides a clear visual representation of these intervals, allowing us to systematically analyze the sign of the first derivative in each. This is a crucial step in determining where the function is increasing or decreasing.

4. Test Values in Each Interval

With the number line divided into intervals by the critical points, we now need to determine the sign of the first derivative, $f'(x)$, in each interval. To do this, we choose test values within each interval and evaluate $f'(x)$ at those points. The sign of $f'(x)$ at the test value will indicate whether the function is increasing (positive $f'(x)$) or decreasing (negative $f'(x)$) in that entire interval.

1. Interval $(0, \frac{\pi}{4})$: Choose a test value, say $x = \frac{\pi}{6}$. Evaluate $f'(\frac{\pi}{6}) = \cos(\frac{\pi}{6}) - \sin(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} - \frac{1}{2}$. Since $\frac{\sqrt{3}}{2} > \frac{1}{2}$, $f'(\frac{\pi}{6}) > 0$. Thus, $f(x)$ is increasing in this interval.
2. Interval $(\frac{\pi}{4}, \frac{5\pi}{4})$: Choose a test value, say $x = \pi$. Evaluate $f'(\pi) = \cos(\pi) - \sin(\pi) = -1 - 0 = -1$. Since $f'(\pi) < 0$, $f(x)$ is decreasing in this interval.
3. Interval $(\frac{5\pi}{4}, 2\pi)$: Choose a test value, say $x = \frac{7\pi}{4}$. Evaluate $f'(\frac{7\pi}{4}) = \cos(\frac{7\pi}{4}) - \sin(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) = \sqrt{2}$. Since $f'(\frac{7\pi}{4}) > 0$, $f(x)$ is increasing in this interval.

By testing these values, we have determined the sign of the first derivative in each interval, which allows us to conclude where the function $f(x)$ is increasing or decreasing.

5. Conclude the Intervals

Based on the sign analysis of the first derivative, $f'(x)$, in each interval, we can now conclude the intervals where the function $f(x) = \sin(x) + \cos(x)$ is increasing or decreasing within the interval $0 < x < 2\pi$. From our test value analysis, we found:

• In the interval $(0, \frac{\pi}{4})$, $f'(x) > 0$, so $f(x)$ is increasing.
• In the interval $(\frac{\pi}{4}, \frac{5\pi}{4})$, $f'(x) < 0$, so $f(x)$ is decreasing.
• In the interval $(\frac{5\pi}{4}, 2\pi)$, $f'(x) > 0$, so $f(x)$ is increasing.

Therefore, the function $f(x) = \sin(x) + \cos(x)$ is increasing on the intervals $(0, \frac{\pi}{4})$ and $(\frac{5\pi}{4}, 2\pi)$, and decreasing on the interval $(\frac{\pi}{4}, \frac{5\pi}{4})$. These conclusions provide a complete picture of the function's behavior within the given domain.

Summary of Increasing and Decreasing Intervals

In summary, by following a systematic approach involving the calculation of the first derivative, finding critical points, and testing intervals, we have successfully identified the intervals where the function $f(x) = \sin(x) + \cos(x)$ is increasing and decreasing within the interval $0 < x < 2\pi$. The key findings are:

• Increasing Intervals: $(0, \frac{\pi}{4})$ and $(\frac{5\pi}{4}, 2\pi)$
• Decreasing Interval: $(\frac{\pi}{4}, \frac{5\pi}{4})$

This analysis provides valuable insights into the behavior of the function and its graph. The intervals of increase and decrease are fundamental concepts in calculus and play a crucial role in understanding the characteristics of functions. This comprehensive exploration demonstrates the method and its application to a trigonometric function, providing a strong foundation for analyzing other functions as well. The ability to determine these intervals is essential for various applications, including optimization problems and curve sketching, highlighting the practical importance of this calculus technique.