The number of point in the rectangle `{(x , y)}-12lt=xlt=12a n d-3lt=ylt=3}` which lie on the curve `y=x+sinx` and at which in the tangent to the curve is parallel to the x-axis is 0 (b) 2 (c) 4 (d) 8

Options

A. 0
B. 2
C. 4
D. 8

Answer

To solve the problem, we need to find the number of points in the rectangle defined by the inequalities \(-12 < x < 12\) and \(-3 < y < 3\) where the curve \(y = x + \sin x\) has a tangent that is parallel to the x-axis. This occurs when the derivative \(dy/dx = 0\).

### Step 1: Find the derivative of the curve
The curve is given by:
\[ y = x + \sin x \]
To find the points where the tangent is parallel to the x-axis, we first compute the derivative:
\[ \frac{dy}{dx} = 1 + \cos x \]

### Step 2: Set the derivative equal to zero
For the tangent to be parallel to the x-axis:
\[ 1 + \cos x = 0 \]
This simplifies to:
\[ \cos x = -1 \]

### Step 3: Solve for \(x\)
The cosine function equals \(-1\) at odd multiples of \(\pi\):
\[ x = (2n + 1)\pi \quad \text{for } n \in \mathbb{Z} \]
This means the solutions are:
\[ x = \pi, -\pi, 3\pi, -3\pi, 5\pi, -5\pi, \ldots \]

### Step 4: Identify valid \(x\) values within the interval
Now we need to find which of these values lie within the interval \(-12 < x < 12\):
- For \(n = 0\): \(x = \pi \approx 3.14\)
- For \(n = -1\): \(x = -\pi \approx -3.14\)
- For \(n = 1\): \(x = 3\pi \approx 9.42\)
- For \(n = -2\): \(x = -3\pi \approx -9.42\)
- For \(n = 2\): \(x = 5\pi \approx 15.71\) (not valid)
- For \(n = -3\): \(x = -5\pi \approx -15.71\) (not valid)

Thus, the valid \(x\) values within the interval \(-12 < x < 12\) are:
\[ x = \pi, -\pi, 3\pi, -3\pi \]

### Step 5: Calculate corresponding \(y\) values
Next, we calculate the corresponding \(y\) values for these \(x\) values:
1. For \(x = \pi\):
\[ y = \pi + \sin(\pi) = \pi + 0 = \pi \approx 3.14 \quad (\text{not valid, as } y < 3) \]
2. For \(x = -\pi\):
\[ y = -\pi + \sin(-\pi) = -\pi + 0 = -\pi \approx -3.14 \quad (\text{not valid, as } y > -3) \]
3. For \(x = 3\pi\):
\[ y = 3\pi + \sin(3\pi) = 3\pi + 0 = 3\pi \quad (\text{not valid, as } y > 3) \]
4. For \(x = -3\pi\):
\[ y = -3\pi + \sin(-3\pi) = -3\pi + 0 = -3\pi \quad (\text{not valid, as } y < -3) \]

### Conclusion
None of the points \((\pi, \pi)\), \((- \pi, -\pi)\), \((3\pi, 3\pi)\), or \((-3\pi, -3\pi)\) lie within the rectangle defined by the inequalities \(-12 < x < 12\) and \(-3 < y < 3\).

Thus, the number of points in the rectangle where the tangent to the curve is parallel to the x-axis is:
\[ \boxed{0} \]