Problem 43 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(41-48\) , consider the function on the interval (0,2 \pi). For each function, (a) find the open interval(s) on which the function is increasing or decreasing, apply the First Derivative Test to identify all relative extrema, and (c) use a graphing utility to confirm your results. $$ f(x)=\sin x+\cos x $$

Step-by-Step Solution

Answer

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 \( (\frac{\pi}{4} , \frac{5\pi}{4}) \). It has a local maximum at \( x = \frac{\pi}{4} \) and a local minimum at \( x = \frac{5\pi}{4} \).

Step 1: Compute the Derivative

Let's start by finding the derivative of the function \( f(x)=\sin x+\cos x \) which gives \( f'(x)=\cos x - \sin x \). The function \( f'(x)\) is the derivative of the original function \( f(x)\)

Step 2: Solve for Critical Points

To find the critical points of the function, let's set the derivative \( f'(x) = 0 \) and solve for \( x \). So, \( \cos x - \sin x = 0 \) implies \( \cos x = \sin x \). The values that satisfy this equation in the interval \((0, 2 \pi) \) are \( \frac{\pi}{4}\) and \( \frac{5\pi}{4}\). These are the critical points

Step 3: Determine Intervals of Increase and Decrease

Let's check the sign of \( f'(x) \) on the intervals determined by the critical points to find where the function is increasing or decreasing. Pick any number from each interval and substitute into \( f'(x) \). If the result is positive, the function is increasing on that interval, and if it's negative, the function is decreasing. Let's choose θ = \( \frac{\pi}{8}, \frac{3\pi}{4}, \frac{7\pi}{4} \) from the corresponding intervals \(( 0 , \frac{\pi}{4} ) , ( \frac{\pi}{4} , \frac{5\pi}{4} ), ( \frac{5\pi}{4} , 2\pi )\). We see that \( f'(\frac{\pi}{8}) > 0\), \( f'(\frac{3\pi}{4}) < 0\) and \( f'(\frac{7\pi}{4}) > 0 \). Therefore, \( f(x) \) is increasing on \(( 0 , \frac{\pi}{4} )\) and \(( \frac{5\pi}{4} , 2\pi )\), and it's decreasing on \( (\frac{\pi}{4} , \frac{5\pi}{4}) \)

Step 4: Apply the First Derivative Test

The First Derivative test states that if \( f'(x) \) changes from positive to negative at a point, then \( f(x) \) has a local maximum there. If \( f'(x) \) changes from negative to positive at a point, then \( f(x) \) has a local minimum there. So, \( f(x) \) has a local maximum at \( x = \frac{\pi}{4} \) and a local minimum at \( x = \frac{5\pi}{4} \).

Step 5: Confirm Results Graphically

The last step is using a graphing utility to graph the original function \( f(x) = \sin x + \cos x \). From the graph, you'll be able to see the local maximum and minimum at \( x = \frac{\pi}{4}, \frac{5\pi}{4} \) respectively and verify the intervals where the function increases and decreases, conforming the findings.

Key Concepts

Critical Points

Critical points are essential in understanding the behavior of a function. They occur where the derivative of the function is zero or undefined, indicating potential locations for local maxima, minima, or points of inflection.
In the given exercise, the critical points were determined by setting the first derivative of the function, \( f'(x) = \cos x - \sin x \), equal to zero. Solving \( \cos x = \sin x \) within the given interval \( (0, 2\pi) \) led us to the critical points \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).
These critical points help us break down the interval into segments that allow further analysis of the function's increase and decrease behavior.

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we analyze the sign of its first derivative over intervals. Specifically, a function is increasing where its derivative is positive and decreasing where its derivative is negative.
In this exercise, following the determination of critical points, the interval \((0, 2\pi)\) is divided into \((0, \frac{\pi}{4}), (\frac{\pi}{4}, \frac{5\pi}{4}),\) and \((\frac{5\pi}{4}, 2\pi)\).

• For \( (0, \frac{\pi}{4}) \), choosing \( \theta = \frac{\pi}{8} \) shows \( f'(\theta) > 0 \), leading to the conclusion that \( f(x) \) increases in this interval.
• At \( (\frac{\pi}{4}, \frac{5\pi}{4}) \), with \( \theta = \frac{3\pi}{4} \), \( f'(\theta) < 0 \), indicating that \( f(x) \) decreases.
• For \( (\frac{5\pi}{4}, 2\pi) \), \( \theta = \frac{7\pi}{4} \) gives \( f'(\theta) > 0 \), showing an increase again.

This step is crucial for understanding the behavior and shape of the function graph in different sections.

Local Extrema

Local extrema are the points on a graph where the function reaches a local maximum or minimum. The First Derivative Test helps identify these points by looking at sign changes of the first derivative around critical points.

• If the first derivative changes from positive to negative at a critical point, there's a local maximum.
• Conversely, if it changes from negative to positive, a local minimum occurs.

In our problem, the test reveals a local maximum at \( x = \frac{\pi}{4} \) and a local minimum at \( x = \frac{5\pi}{4} \). Recognizing these changes in behavior helps visualize how the function peaks and dips over its domain.

Graphing Utility

A graphing utility is a helpful tool to visually confirm analytical findings from derivative tests by plotting the function.
In this scenario, plotting \( f(x) = \sin x + \cos x \) using a graphing utility allows us to observe:

• The intervals of increase and decrease, validating our previous calculations.
• The local maximum and minimum at \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \), respectively.

The visual representation from the graph can strengthen understanding by connecting analytical methods with visual observation. It serves as an integral part of verifying results, ensuring both the theoretical and graphical analysis align.