Find the value of `a`, if the equation `x-sinx=a` has a unique root in `[-(pi)/2,pi/2]`

Options

A. `a in [-(pi)/(2)+1,oo)`
B. ` a in (-oo,(pi)/(2)-1]`
C. ` a in [1-(pi)/(2),(pi)/(2)-1]`
D. ` a in R-(1-(pi)/(2),(pi)/(2)-1)`

Answer

To find the value of \( a \) such that the equation \( x - \sin x = a \) has a unique root in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), we can follow these steps:

### Step 1: Define the function
We define the function:
\[ f(x) = x - \sin x - a \]

### Step 2: Find the derivative
Next, we find the derivative of \( f(x) \):
\[ f'(x) = 1 - \cos x \]
Using the identity \( 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \), we can see that:
\[ f'(x) = 2 \sin^2\left(\frac{x}{2}\right) \]
Since \( \sin^2\left(\frac{x}{2}\right) \geq 0 \) for all \( x \), it follows that:
\[ f'(x) \geq 0 \quad \text{for all } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]

### Step 3: Determine the nature of \( f(x) \)
Since \( f'(x) \) is non-negative, \( f(x) \) is a non-decreasing function. To have a unique root, \( f(x) \) must change sign exactly once in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).

### Step 4: Evaluate \( f(x) \) at the endpoints
We evaluate \( f(x) \) at the endpoints of the interval:
1. At \( x = -\frac{\pi}{2} \):
\[ f\left(-\frac{\pi}{2}\right) = -\frac{\pi}{2} - \sin\left(-\frac{\pi}{2}\right) - a = -\frac{\pi}{2} + 1 - a \]
We require:
\[ f\left(-\frac{\pi}{2}\right) \leq 0 \implies -\frac{\pi}{2} + 1 - a \leq 0 \implies a \geq 1 - \frac{\pi}{2} \]

2. At \( x = \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2}\right) = \frac{\pi}{2} - \sin\left(\frac{\pi}{2}\right) - a = \frac{\pi}{2} - 1 - a \]
We require:
\[ f\left(\frac{\pi}{2}\right) \geq 0 \implies \frac{\pi}{2} - 1 - a \geq 0 \implies a \leq \frac{\pi}{2} - 1 \]

### Step 5: Combine the inequalities
From the evaluations, we have:
\[ 1 - \frac{\pi}{2} \leq a \leq \frac{\pi}{2} - 1 \]

### Step 6: Final result
Thus, the values of \( a \) for which the equation \( x - \sin x = a \) has a unique root in the interval \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) are:
\[ a \in \left[1 - \frac{\pi}{2}, \frac{\pi}{2} - 1\right] \]

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