Let `f(x)=max{tanx, cotx}`. Then the number of roots of the equation `f(x)=(1)/(2)" in "(0, 2pi)` is

Options

A. 0
B. 1
C. 2
D. 4

Answer

To solve the problem, we need to analyze the function \( f(x) = \max(\tan x, \cot x) \) and determine the number of roots of the equation \( f(x) = \frac{1}{2} \) in the interval \( (0, 2\pi) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
- The function \( f(x) = \max(\tan x, \cot x) \) means that for any value of \( x \), \( f(x) \) will take the larger value between \( \tan x \) and \( \cot x \).
- We need to find out in which intervals \( \tan x \) is greater than \( \cot x \) and vice versa.

2. **Finding Intervals**:
- The functions \( \tan x \) and \( \cot x \) are equal when \( \tan x = \cot x \), which occurs at \( x = \frac{\pi}{4} + n\pi \) for integer \( n \).
- In the interval \( (0, 2\pi) \), this gives us the points \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \).

3. **Analyzing Intervals**:
- For \( x \in (0, \frac{\pi}{4}) \):
- \( \tan x < \cot x \) ⇒ \( f(x) = \cot x \)
- For \( x \in (\frac{\pi}{4}, \frac{5\pi}{4}) \):
- \( \tan x > \cot x \) ⇒ \( f(x) = \tan x \)
- For \( x \in (\frac{5\pi}{4}, 2\pi) \):
- \( \tan x < \cot x \) ⇒ \( f(x) = \cot x \)

4. **Setting Up the Equation**:
- We need to solve \( f(x) = \frac{1}{2} \).
- In the intervals where \( f(x) = \cot x \):
- \( \cot x = \frac{1}{2} \) implies \( \tan x = 2 \).
- In the intervals where \( f(x) = \tan x \):
- \( \tan x = \frac{1}{2} \).

5. **Finding Roots**:
- **For \( \tan x = \frac{1}{2} \)**:
- The general solution is \( x = \tan^{-1}(\frac{1}{2}) + n\pi \).
- In the interval \( (0, 2\pi) \), this gives us two solutions.
- **For \( \cot x = \frac{1}{2} \)**:
- The general solution is \( x = \cot^{-1}(\frac{1}{2}) + n\pi \).
- In the interval \( (0, 2\pi) \), this also gives us two solutions.

6. **Counting Unique Roots**:
- We have two solutions from \( \tan x = \frac{1}{2} \) and two solutions from \( \cot x = \frac{1}{2} \).
- However, we need to check if any of these solutions overlap.
- Since \( \tan x \) and \( \cot x \) are periodic and have different ranges, the solutions will not overlap.

### Conclusion:
Thus, the total number of roots of the equation \( f(x) = \frac{1}{2} \) in the interval \( (0, 2\pi) \) is **4**.

### Final Answer:
The number of roots of the equation \( f(x) = \frac{1}{2} \) in \( (0, 2\pi) \) is **4**.