The total number of solution(s) of the equation `2x+3 tanx=(5pi)/(2)` in ` x in [0, 2pi]` is/are equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the total number of solutions for the equation \(2x + 3 \tan x = \frac{5\pi}{2}\) in the interval \(x \in [0, 2\pi]\), we can follow these steps:

### Step-by-Step Solution:

1. **Rearranging the Equation:**
Start by rewriting the equation in a more manageable form:
\[ 3 \tan x = \frac{5\pi}{2} - 2x \]
This gives us:
\[ \tan x = \frac{5\pi}{6} - \frac{2}{3}x \]

2. **Identifying Functions:**
Let \(y_1 = \tan x\) and \(y_2 = \frac{5\pi}{6} - \frac{2}{3}x\). We will analyze the intersections of these two functions to find the solutions.

3. **Domain Consideration:**
The function \(\tan x\) is periodic with vertical asymptotes at \(x = \frac{\pi}{2} + n\pi\) for integers \(n\). Within the interval \([0, 2\pi]\), the asymptotes occur at:
- \(x = \frac{\pi}{2}\)
- \(x = \frac{3\pi}{2}\)

4. **Graphing the Functions:**
- The graph of \(y_1 = \tan x\) will have vertical asymptotes at \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\).
- The graph of \(y_2 = \frac{5\pi}{6} - \frac{2}{3}x\) is a straight line with a y-intercept of \(\frac{5\pi}{6}\) and a slope of \(-\frac{2}{3}\).

5. **Finding Intersections:**
We need to determine how many times the line \(y_2\) intersects the curve \(y_1\) within the intervals:
- From \(0\) to \(\frac{\pi}{2}\)
- From \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\)
- From \(\frac{3\pi}{2}\) to \(2\pi\)

6. **Analyzing Each Interval:**
- **Interval [0, \(\frac{\pi}{2}\)]:** The line starts above the x-axis and decreases, while \(\tan x\) starts at \(0\) and increases to \(+\infty\). There is **1 intersection**.
- **Interval [\(\frac{\pi}{2}\), \(\frac{3\pi}{2}\)]:** The line continues to decrease and \(\tan x\) goes from \(+\infty\) to \(-\infty\). There is **1 intersection**.
- **Interval [\(\frac{3\pi}{2}\), \(2\pi\)]:** The line again decreases and \(\tan x\) goes from \(+\infty\) to \(0\). There is **1 intersection**.

7. **Total Number of Solutions:**
Adding the intersections from each interval gives us a total of:
\[ 1 + 1 + 1 = 3 \]

### Final Answer:
The total number of solutions of the equation \(2x + 3 \tan x = \frac{5\pi}{2}\) in the interval \(x \in [0, 2\pi]\) is **3**.