Problem 36 - Calculus Early Transcendentals

Question

Analyze the trigonometric function f over the specified interval, stating where f is increasing, decreasing, concave up, and concave down, and stating the x-coordinates of all inflection points. Confirm that your results are consistent with the graph of f generated with a graphing utility. \(f(x)=2 x+\cot x ;(0, \pi)\)

Step-by-Step Solution

Answer

f is increasing on \((0, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \pi)\), decreasing on \((\frac{\pi}{4}, \frac{3\pi}{4})\); concave up on \((0, \frac{\pi}{2})\), concave down on \((\frac{\pi}{2}, \pi)\); inflection point at \(x = \frac{\pi}{2}\).

Step 1: Determine the First Derivative

The first step is to find the first derivative of the function \(f(x) = 2x + \cot x\). Using the derivative rules, the derivative of \(2x\) is \(2\) and the derivative of \(\cot x\) is \(-\csc^2 x\). Therefore, \(f'(x) = 2 - \csc^2 x\).

Step 2: Identify Increasing/Decreasing Intervals

An increasing function has a positive derivative, whereas a decreasing function has a negative derivative. Set \(f'(x) = 2 - \csc^2 x = 0\) to find critical points. Solving gives \(\csc^2 x = 2\), or equivalently \(\sin^2 x = \frac{1}{2}\), giving \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\) in the interval \((0, \pi)\). Thus, analyze intervals \( (0, \frac{\pi}{4})\), \((\frac{\pi}{4}, \frac{3\pi}{4})\), and \((\frac{3\pi}{4}, \pi)\) to determine where \(f'(x)\) is positive or negative by plugging in test points into \(f'(x)\).

Step 3: Determine Concavity Using the Second Derivative

Calculate the second derivative \(f''(x)\), where \(f'(x) = 2 - \csc^2 x\). The derivative of \(-\csc^2 x\) is \(2\csc^3 x \cot x\), so \(f''(x) = 2\csc^3 x \cot x\). Using \(f''(x) > 0\) indicates concave up and \(f''(x) < 0\) indicates concave down. Analyze where \(f''(x)\) changes sign to find inflection points.

Step 4: Find Inflection Points

Identify where \(f''(x) = 2\csc^3 x \cot x\) changes sign within \((0, \pi)\). Notice that \(\csc^3 x \cot x = 0\) when \(\cot x = 0\), which occurs at \(x = \frac{\pi}{2}\). Therefore, there is a potential inflection point at this value. Confirm by sign change analysis across \(x = \frac{\pi}{2}\).

Step 5: Confirm with Graphing Utility

Generate a graph of \(f(x) = 2x + \cot x\) over \((0, \pi)\) using a graphing utility. Verify that the function matches the identified intervals for increasing/decreasing and concavity and locate the inflection point visually to confirm.

Key Concepts

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we need to investigate the sign of its first derivative, denoted as \(f'(x)\). For the function \(f(x) = 2x + \cot x\), the first derivative is \(f'(x) = 2 - \csc^2 x\).

Analyzing this derivative, a function is considered increasing in intervals where \(f'(x) > 0\), and decreasing where \(f'(x) < 0\).

In our scenario, solving \(2 - \csc^2 x = 0\) gives critical points at \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\). These points divide the interval \((0, \pi)\) into sections:

• \((0, \frac{\pi}{4})\)
• \((\frac{\pi}{4}, \frac{3\pi}{4})\)
• \((\frac{3\pi}{4}, \pi)\)

By evaluating \(f'(x)\) in each interval, we notice:

• The function is increasing in \((0, \frac{\pi}{4})\) and \((\frac{3\pi}{4}, \pi)\) because \(f'(x) > 0\).
• It is decreasing in \((\frac{\pi}{4}, \frac{3\pi}{4})\) where \(f'(x) < 0\).

Understanding these intervals helps us sketch the function's behavior across its domain.

Concavity and Inflection Points

Concavity describes how a function curves up or down and is determined using the second derivative, \(f''(x)\). For \(f(x) = 2x + \cot x\), the second derivative is \(f''(x) = 2\csc^3 x \cot x\).

To identify concavity:

• The function is concave up where \(f''(x) > 0\).
• It is concave down where \(f''(x) < 0\).

For inflection points, we look for values of \(x\) where \(f''(x)\) changes signs. At \(x = \frac{\pi}{2}\), the derivative becomes zero or undefined, marking a possible inflection point.

Around \(x = \frac{\pi}{2}\), checking the sign change of \(f''(x)\) confirms the inflection point. This point indicates a transition in the function's curvature, essential for understanding how the graph looks.

Identifying these characteristics gives more insight into how the function behaves across its domain.

Graphing Utilities

Visualizing a function using graphing utilities can significantly aid in verifying analytical results. These tools can plot complex functions over a specified interval, like \((0, \pi)\) for \(f(x) = 2x + \cot x\).

Graphing utilities allow for:

• Viewing where the function increases or decreases.
• Observing concave up and concave down sections.
• Pinning down inflection points visually.

To generate the graph, input the equation into the graphing software and adjust the view to focus on the open interval \((0, \pi)\). You should see the increase in \((0, \frac{\pi}{4})\) and \((\frac{3\pi}{4}, \pi)\), along with a decreasing trend in \((\frac{\pi}{4}, \frac{3\pi}{4})\).

Additionally, look for changes in curvature, confirming our analytical finding of an inflection point at \(x = \frac{\pi}{2}\). The graphical representation assists in a deeper understanding and validation of the function's properties.