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 the interval \([-\frac{\pi}{2}, \frac{3\pi}{2}]\), we can follow these steps:

### Step 1: Understand the Functions
We need to analyze the two functions:
1. \(y = |\sin x|\)
2. \(y = -x^2\)

### Step 2: Analyze the Function \(y = |\sin x|\)
The function \(|\sin x|\) oscillates between 0 and 1. Specifically, within the interval \([-\frac{\pi}{2}, \frac{3\pi}{2}]\):
- At \(x = -\frac{\pi}{2}\), \(|\sin(-\frac{\pi}{2})| = 1\)
- At \(x = 0\), \(|\sin(0)| = 0\)
- At \(x = \frac{\pi}{2}\), \(|\sin(\frac{\pi}{2})| = 1\)
- At \(x = \pi\), \(|\sin(\pi)| = 0\)
- At \(x = \frac{3\pi}{2}\), \(|\sin(\frac{3\pi}{2})| = 1\)

### Step 3: Analyze the Function \(y = -x^2\)
The function \(-x^2\) is a downward-opening parabola:
- At \(x = 0\), \(-x^2 = 0\)
- At \(x = 1\) or \(x = -1\), \(-x^2 = -1\)
- As \(x\) moves away from 0, \(-x^2\) decreases without bound.

### Step 4: Sketch the Graphs
1. **Graph of \(|\sin x|\)**: This will oscillate between 0 and 1.
2. **Graph of \(-x^2\)**: This will start at 0 when \(x = 0\) and go downwards, intersecting the y-axis at \(0\) and reaching \(-1\) at \(x = 1\) and \(x = -1\).

### Step 5: Find Intersections
We need to find the points where these two graphs intersect:
- The function \(|\sin x|\) can only take values between 0 and 1.
- The function \(-x^2\) is always less than or equal to 0 for \(x \neq 0\) and is 0 at \(x = 0\).

### Step 6: Conclusion
The only point where \(|\sin x| = -x^2\) can hold true is at \(x = 0\):
- At \(x = 0\), \(|\sin(0)| = 0\) and \(-0^2 = 0\).

Thus, there is exactly **one solution** in the interval \([-\frac{\pi}{2}, \frac{3\pi}{2}]\).

### Final Answer
The number of solutions is **1**.

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