Question

The value of b for which the function f(x) = sinx - bx + C, where b and e are constants is decreasing for \(x \in R\) is given by

• A. \(b < 1\)
• B. \(b \geq 0\)
• C. \(b > 1\)
• D. \(b \leq 1\)

Answer 1

The Correct Option is C

For the function \( f(x) = \sin x - bx + C \) to be decreasing for \( x \in \mathbb{R} \), its derivative \( f'(x) \) must be non-positive for all \( x \). Compute the derivative:

\( f'(x) = \cos x - b \)

For \( f(x) \) to be decreasing, we need:

\( \cos x - b \leq 0 \) for all \( x \)

This implies:

\( \cos x \leq b \)

Since the range of \( \cos x \) is \([-1, 1]\), to satisfy \( \cos x \leq b \) for all \( x \), \( b \) must be greater than or equal to the maximum value of \(\cos x\):

\( b \geq 1 \)

Hence, for the function to be strictly decreasing, \( b \) must be:

\( b > 1 \)