The number of points in `(-oo,oo),` for which `x^2-xsinx-cosx=0,` is 6 (b) 4 (c) 2 (d) 0

Options

A. 6
B. 4
C. 2
D. 0

Explanation

To solve the equation \( x^2 - x \sin x - \cos x = 0 \) and find the number of points in the interval \((-∞, ∞)\) for which this equation holds true, we will follow these steps:

### Step 1: Define the Function
Let \( f(x) = x^2 - x \sin x - \cos x \).

### Step 2: Find the Derivative
To analyze the behavior of the function, we need to find its derivative:
\[ f'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(x \sin x) - \frac{d}{dx}(\cos x) \]
Using the product rule on \( x \sin x \):
\[ f'(x) = 2x - (\sin x + x \cos x) + \sin x \]
This simplifies to:
\[ f'(x) = 2x - x \cos x \]
Factoring out \( x \):
\[ f'(x) = x(2 - \cos x) \]

### Step 3: Analyze the Derivative
Now we analyze \( f'(x) \):
- The term \( 2 - \cos x \) is always positive because the maximum value of \( \cos x \) is 1. Thus, \( 2 - \cos x \geq 1 \).
- Therefore, \( f'(x) \) is:
- Positive when \( x > 0 \) (since \( x > 0 \) and \( 2 - \cos x > 0 \)).
- Negative when \( x < 0 \) (since \( x < 0 \) and \( 2 - \cos x > 0 \)).
- Zero when \( x = 0 \).

This indicates that \( f(x) \) is decreasing for \( x < 0 \) and increasing for \( x > 0 \).

### Step 4: Evaluate the Function at Key Points
Next, we evaluate \( f(x) \) at \( x = 0 \):
\[ f(0) = 0^2 - 0 \cdot \sin(0) - \cos(0) = -1 \]
Now, let's check the limits as \( x \) approaches positive and negative infinity:
- As \( x \to \infty \), \( f(x) \to \infty \) (since \( x^2 \) dominates).
- As \( x \to -\infty \), \( f(x) \to \infty \) (since \( x^2 \) still dominates).

### Step 5: Determine the Number of Roots
Since \( f(x) \) is decreasing for \( x < 0 \) and \( f(0) < 0 \), \( f(x) \) must cross the x-axis at least once in the interval \( (-\infty, 0) \).

For \( x > 0 \), since \( f(x) \to \infty \) as \( x \to \infty \) and \( f(0) < 0 \), there must be at least one root in the interval \( (0, \infty) \).

### Conclusion
Thus, we find that there are two points where \( f(x) = 0 \), one in \( (-\infty, 0) \) and one in \( (0, \infty) \).

The number of points in \( (-\infty, \infty) \) for which \( x^2 - x \sin x - \cos x = 0 \) is **2**.

### Answer
The correct option is (c) 2.