Number of solutions (s) of in `|sin x|=-x ^(2)` if `x in [-(pi)/(2), (3pi)/(2)]` is/are:

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the number of solutions to the equation \( |\sin x| = -x^2 \) for \( x \in \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right] \), we can follow these steps:

### Step 1: Understand the Functions
We need to analyze the functions \( |\sin x| \) and \( -x^2 \). The left side \( |\sin x| \) is always non-negative since it represents the absolute value of sine. The right side, \( -x^2 \), is always non-positive because it is the negative of a square.

### Step 2: Set Up the Equation
Since \( |\sin x| \) is non-negative and \( -x^2 \) is non-positive, the only way for these two functions to be equal is if both sides are equal to zero. Therefore, we need to solve:
\[ |\sin x| = 0 \quad \text{and} \quad -x^2 = 0 \]

### Step 3: Solve for \( |\sin x| = 0 \)
The equation \( |\sin x| = 0 \) occurs when:
\[ \sin x = 0 \]
This happens at:
\[ x = n\pi \quad \text{for integers } n \]
Within the interval \( \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right] \), the values of \( n \) that satisfy this are:
- \( n = 0 \) gives \( x = 0 \)
- \( n = 1 \) gives \( x = \pi \)

### Step 4: Solve for \( -x^2 = 0 \)
The equation \( -x^2 = 0 \) gives:
\[ x^2 = 0 \implies x = 0 \]

### Step 5: Identify the Solutions
From the analysis:
- The solutions from \( |\sin x| = 0 \) are \( x = 0 \) and \( x = \pi \).
- The solution from \( -x^2 = 0 \) is \( x = 0 \).

### Step 6: Count Unique Solutions
The unique solutions in the interval \( \left[-\frac{\pi}{2}, \frac{3\pi}{2}\right] \) are:
- \( x = 0 \)
- \( x = \pi \)

Thus, there are **2 unique solutions**.

### Final Answer
The number of solutions \( s \) is \( 2 \).

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