The value of b for which the function `f (x)=sin x-bx+c `is decreasing in the interval `(- oo,oo)` is given by

Options

A. ` b lt 1`
B. ` b ge 1`
C. ` b gt 1`
D. ` b le 1`

Answer

To determine the value of \( b \) for which the function \( f(x) = \sin x - bx + c \) is decreasing in the interval \( (-\infty, \infty) \), we follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \):
\[ f'(x) = \frac{d}{dx}(\sin x - bx + c) \]
Using the rules of differentiation, we get:
\[ f'(x) = \cos x - b \]

### Step 2: Set the derivative less than or equal to zero
For the function \( f(x) \) to be decreasing, we need:
\[ f'(x) \leq 0 \]
This means:
\[ \cos x - b \leq 0 \]
Rearranging this inequality gives:
\[ \cos x \leq b \]

### Step 3: Analyze the range of \( \cos x \)
The cosine function oscillates between -1 and 1. Therefore, the range of \( \cos x \) is:
\[ -1 \leq \cos x \leq 1 \]

### Step 4: Determine the conditions for \( b \)
From the inequality \( \cos x \leq b \), we need to ensure that \( b \) is greater than or equal to the maximum value of \( \cos x \). Since the maximum value of \( \cos x \) is 1, we have:
\[ b \geq 1 \]

### Conclusion
Thus, the value of \( b \) for which the function \( f(x) = \sin x - bx + c \) is decreasing in the interval \( (-\infty, \infty) \) is:
\[ b \geq 1 \]