To solve the problem, we need to analyze the function \( f(x) = \max\{\tan x, \cot x\} \) and find the number of roots of the equation \( f(x) = \frac{1}{2} \) in the interval \( (0, 2\pi) \).
### Step-by-Step Solution:
1. **Understand the Functions**:
- The function \( f(x) \) is defined as the maximum of \( \tan x \) and \( \cot x \).
- We need to analyze both \( \tan x \) and \( \cot x \) over the interval \( (0, 2\pi) \).
2. **Identify Key Points**:
- The functions \( \tan x \) and \( \cot x \) have vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) respectively.
- The points where \( \tan x = \cot x \) can be found by solving \( \tan x = \cot x \), which simplifies to \( \tan^2 x = 1 \). This gives us \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).
3. **Graph the Functions**:
- Sketch the graphs of \( \tan x \) and \( \cot x \) in the interval \( (0, 2\pi) \):
- From \( 0 \) to \( \frac{\pi}{2} \), \( \tan x \) increases from \( 0 \) to \( +\infty \) and \( \cot x \) decreases from \( +\infty \) to \( 0 \).
- From \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), \( \tan x \) decreases from \( -\infty \) to \( 0 \) and \( \cot x \) increases from \( 0 \) to \( +\infty \).
- From \( \frac{3\pi}{2} \) to \( 2\pi \), \( \tan x \) increases from \( 0 \) to \( +\infty \) and \( \cot x \) decreases from \( +\infty \) to \( 0 \).
4. **Determine the Maximum Function**:
- In the interval \( (0, \frac{\pi}{2}) \), \( f(x) = \tan x \).
- In the interval \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), \( f(x) = \cot x \).
- In the interval \( (\frac{3\pi}{2}, 2\pi) \), \( f(x) = \tan x \).
5. **Find the Intersection with \( y = \frac{1}{2} \)**:
- In \( (0, \frac{\pi}{2}) \), we set \( \tan x = \frac{1}{2} \). This will have one solution since \( \tan x \) is continuous and increasing.
- In \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), we set \( \cot x = \frac{1}{2} \). This will also have one solution since \( \cot x \) is continuous and decreasing.
- In \( (\frac{3\pi}{2}, 2\pi) \), we set \( \tan x = \frac{1}{2} \). This will have one solution as well.
6. **Count the Total Roots**:
- From the analysis, we find:
- One root in \( (0, \frac{\pi}{2}) \)
- One root in \( (\frac{\pi}{2}, \frac{3\pi}{2}) \)
- One root in \( (\frac{3\pi}{2}, 2\pi) \)
- Thus, the total number of roots of the equation \( f(x) = \frac{1}{2} \) in the interval \( (0, 2\pi) \) is **3**.
### Conclusion:
The number of roots of the equation \( f(x) = \frac{1}{2} \) in the interval \( (0, 2\pi) \) is **3**.
