https://file.aiquickdraw.com/mathbot/file/17276680606070eo7j2tm.jpg

To identify where the function \( f(x) = x + 2 \sin(x) \) is increasing or decreasing, we need to:

1. **Find the derivative of \( f(x) \):**
   - \( f'(x) = \frac{d}{dx} \left( x + 2 \sin(x) \right) \)
   - This gives: \( f'(x) = 1 + 2 \cos(x) \)

2. **Determine where \( f'(x) > 0 \) and \( f'(x) < 0 \):**
   - To find when the function is increasing, we solve \( f'(x) > 0 \).
   - To find when the function is decreasing, we solve \( f'(x) < 0 \).

Let's start by analyzing the critical points:
- \( f'(x) = 0 \Rightarrow 1 + 2 \cos(x) = 0 \Rightarrow \cos(x) = -\frac{1}{2} \)
- This occurs at \( x = \frac{2\pi}{3} \) and \( x = \frac{4\pi}{3} \) in the interval \( (0, 2\pi) \).

Now, we check the sign of \( f'(x) \) in each of the three intervals formed by these critical points:
- \( (0, \frac{2\pi}{3}) \)
- \( (\frac{2\pi}{3}, \frac{4\pi}{3}) \)
- \( (\frac{4\pi}{3}, 2\pi) \)

Next, we will test the sign of \( f'(x) \) in these intervals:

1. For \( 0 < x < \frac{2\pi}{3} \), we choose a test point like \( x = \frac{\pi}{2} \):
   - \( f'(\frac{\pi}{2}) = 1 + 2 \cdot 0 = 1 \) (positive, so increasing in this interval).

2. For \( \frac{2\pi}{3} < x < \frac{4\pi}{3} \), we choose a test point like \( x = \pi \):
   - \( f'(\pi) = 1 + 2(-1) = -1 \) (negative, so decreasing in this interval).

3. For \( \frac{4\pi}{3} < x < 2\pi \), we choose a test point like \( x = \frac{3\pi}{2} \):
   - \( f'(\frac{3\pi}{2}) = 1 + 2 \cdot 0 = 1 \) (positive, so increasing in this interval).

### Final Answer:
- **Increasing intervals**: \( (0, \frac{2\pi}{3}) \cup (\frac{4\pi}{3}, 2\pi) \)
- **Decreasing intervals**: \( (\frac{2\pi}{3}, \frac{4\pi}{3}) \)

Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:
1. How would the behavior change if the function was \( f(x) = x + \sin(x) \)?
2. What role does the amplitude of the sine function play in increasing or decreasing intervals?
3. Can you find the points where the function has a local maximum or minimum?
4. How would you determine concavity and points of inflection for this function?
5. What happens to the increasing and decreasing intervals if the domain changes?

**Tip**: When solving for critical points, always check both the derivative's sign changes and any potential boundary points in the given domain!

Identify the open intervals on which the function f(x) = x + 2sin(x) is increasing or decreasing (0 < x < 2π). Enter your answers using interval notation.

Learn how to find the open intervals where the function f(x) = x + 2sin(x) is increasing or decreasing over the interval (0, 2π). Includes derivative calculations, critical points, and interval testing for positive and negative signs of f'(x).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation