The number of solutions of the equation `2 "tan" x + x = (12 pi)/(5) " in the interval " [0, 2 pi]`, is

Options

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

Explanation

To find the number of solutions of the equation \( 2 \tan x + x = \frac{12 \pi}{5} \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 2 \tan x + x = \frac{12 \pi}{5} \]
Rearranging gives:
\[ 2 \tan x = \frac{12 \pi}{5} - x \]
Thus,
\[ \tan x = \frac{12 \pi}{10} - \frac{x}{2} = \frac{6 \pi}{5} - \frac{x}{2} \]

**Hint:** Rearranging the equation helps isolate the trigonometric function.

### Step 2: Setting Up the Function
Let \( y = \tan x \). We can express the equation as:
\[ y = \frac{6 \pi}{5} - \frac{x}{2} \]
This is a linear equation in \( x \) with slope \( -\frac{1}{2} \) and y-intercept \( \frac{6 \pi}{5} \).

**Hint:** Identifying the linear equation allows us to analyze its graph alongside the tangent function.

### Step 3: Analyzing the Graphs
We need to analyze the graphs of \( y = \tan x \) and \( y = \frac{6 \pi}{5} - \frac{x}{2} \) over the interval \( [0, 2\pi] \).

- The function \( \tan x \) has vertical asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
- The line \( y = \frac{6 \pi}{5} - \frac{x}{2} \) will intersect the y-axis at \( \frac{6 \pi}{5} \) and has a negative slope.

**Hint:** Sketching the graphs will help visualize the intersection points.

### Step 4: Finding Intersection Points
1. **At \( x = 0 \)**:
\[ \tan(0) = 0 \quad \text{and} \quad y = \frac{6 \pi}{5} \quad \text{(line)} \]
The line is above the tangent curve.

2. **At \( x = \frac{\pi}{2} \)**:
The tangent function approaches \( +\infty \).

3. **At \( x = \frac{3\pi}{2} \)**:
The tangent function approaches \( -\infty \).

4. **At \( x = 2\pi \)**:
\[ \tan(2\pi) = 0 \quad \text{and} \quad y = \frac{6 \pi}{5} - \frac{2\pi}{2} = \frac{6 \pi}{5} - \pi = \frac{\pi}{5} \quad \text{(line)} \]
The line is above the tangent curve.

**Hint:** Check the values of the functions at key points to determine where they intersect.

### Step 5: Conclusion on Number of Solutions
From the analysis:
- The line intersects the tangent curve twice in the interval \( [0, 2\pi] \) (once before \( x = \frac{\pi}{2} \) and once after \( x = \frac{3\pi}{2} \)).
- Therefore, the number of solutions to the equation \( 2 \tan x + x = \frac{12 \pi}{5} \) in the interval \( [0, 2\pi] \) is **2**.

**Final Answer:**
The number of solutions is \( \boxed{2} \).