Question

The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when

• A. \( k > 1 \)
• B. \( k \geq 1 \)
• C. \( k < 1 \)
• D. \( k \leq 1 \)
• E. \( k < -1 \)

Hint:

For “always” conditions, always test using maximum/minimum values of trigonometric functions.

Answer 1

The Correct Option is B

Concept:
• A function is said to be monotonically decreasing in an interval if its derivative is less than or equal to zero for all values in that interval.
• For always decreasing (i.e., for all real \(x\)): \[ f'(x) \leq 0 \quad \forall x \in \mathbb{R} \]

Step 1: Write the given function clearly.
\[ f(x) = \sin x - kx - c \] Here:
• \( \sin x \) is a periodic function
• \( -kx \) is linear
• \( c \) is a constant (does not affect slope)

Step 2: Differentiate the function.
Differentiate term-by-term: \[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(-kx) = -k \] \[ \frac{d}{dx}(-c) = 0 \] Thus, derivative becomes: \[ f'(x) = \cos x - k \]

Step 3: Apply condition for decreasing function.
For the function to be decreasing for all values of \(x\): \[ f'(x) \leq 0 \] \[ \Rightarrow \cos x - k \leq 0 \] \[ \Rightarrow \cos x \leq k \]

Step 4: Analyze the inequality in detail.
We need: \[ \cos x \leq k \quad \text{for every real } x \] Now recall the range of cosine function: \[ -1 \leq \cos x \leq 1 \] This means:
• The maximum possible value of \( \cos x \) is \(1\)
• The minimum possible value of \( \cos x \) is \(-1\)

Step 5: Find the strict condition for inequality.
Since \( \cos x \) can take values up to \(1\), the inequality must hold even at its maximum. So: \[ 1 \leq k \] This ensures: \[ \cos x \leq 1 \leq k \] Thus: \[ \cos x \leq k \quad \forall x \]

Step 6: Check what happens if condition is violated.
If \( k < 1 \), consider \( x = 0 \): \[ \cos 0 = 1 \] Then: \[ \cos x - k = 1 - k > 0 \] So: \[ f'(x) > 0 \] This means:
• Function increases at some points
• Hence NOT always decreasing Thus such values of \(k\) are not valid.

Step 7: Final conclusion.
The condition for the function to be decreasing for all \(x\) is: \[ k \geq 1 \]

Step 8: Final Answer.
\[ \boxed{k \geq 1} \]