The number of roots of the equation `x +2"tan"x = (pi)/(2)` in the interval` [0, 2 pi]`, is

Options

A. 1
B. 2
C. 3
D. 6

Answer

To find the number of roots of the equation \( x + 2\tan x = \frac{\pi}{2} \) in the interval \( [0, 2\pi] \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x + 2\tan x = \frac{\pi}{2} \]
We can rearrange it to isolate \( \tan x \):
\[ 2\tan x = \frac{\pi}{2} - x \]
Dividing both sides by 2 gives:
\[ \tan x = \frac{\pi}{4} - \frac{x}{2} \]

### Step 2: Analyzing the Functions
Now we will analyze the two functions involved:
1. \( y_1 = \tan x \)
2. \( y_2 = \frac{\pi}{4} - \frac{x}{2} \)

### Step 3: Finding the Domain of \( \tan x \)
The function \( \tan x \) is defined and continuous in the intervals where it does not have vertical asymptotes. In the interval \( [0, 2\pi] \), \( \tan x \) is undefined at:
- \( x = \frac{\pi}{2} \)
- \( x = \frac{3\pi}{2} \)

Thus, we will analyze the intervals:
- \( [0, \frac{\pi}{2}) \)
- \( (\frac{\pi}{2}, \frac{3\pi}{2}) \)
- \( (\frac{3\pi}{2}, 2\pi] \)

### Step 4: Behavior of \( y_1 = \tan x \)
- In \( [0, \frac{\pi}{2}) \), \( \tan x \) increases from \( 0 \) to \( +\infty \).
- In \( (\frac{\pi}{2}, \frac{3\pi}{2}) \), \( \tan x \) decreases from \( -\infty \) to \( 0 \).
- In \( (\frac{3\pi}{2}, 2\pi] \), \( \tan x \) increases from \( 0 \) to \( +\infty \).

### Step 5: Behavior of \( y_2 = \frac{\pi}{4} - \frac{x}{2} \)
The function \( y_2 \) is a linear function that decreases from \( \frac{\pi}{4} \) at \( x = 0 \) to \( -\frac{\pi}{4} \) at \( x = 2\pi \).

### Step 6: Finding Intersections
Now we will find the points where \( y_1 \) and \( y_2 \) intersect in each interval:
1. **In \( [0, \frac{\pi}{2}) \)**:
- \( \tan x \) starts at \( 0 \) and goes to \( +\infty \).
- \( y_2 \) starts at \( \frac{\pi}{4} \) and goes to \( 0 \).
- There is **1 intersection**.

2. **In \( (\frac{\pi}{2}, \frac{3\pi}{2}) \)**:
- \( \tan x \) goes from \( -\infty \) to \( 0 \).
- \( y_2 \) goes from \( 0 \) to \( -\frac{\pi}{4} \).
- There is **1 intersection**.

3. **In \( (\frac{3\pi}{2}, 2\pi] \)**:
- \( \tan x \) goes from \( 0 \) to \( +\infty \).
- \( y_2 \) goes from \( -\frac{\pi}{4} \) to \( -\frac{\pi}{4} \).
- There is **1 intersection**.

### Conclusion
Adding the intersections from all intervals, we find a total of:
\[ 1 + 1 + 1 = 3 \]
Thus, the number of roots of the equation \( x + 2\tan x = \frac{\pi}{2} \) in the interval \( [0, 2\pi] \) is **3**.