Question

The function \( f(x) = kx - \sin x \) is strictly increasing for:

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

Hint:

For strict monotonicity, check the sign of the derivative over the entire domain.

Answer 1

The Correct Option is A

Step 1: Find the derivative
The derivative of \( f(x) \) is:
\[ f'(x) = k - \cos x. \]
Step 2: Condition for increasing function
For \( f(x) \) to be strictly increasing: \[ f'(x)>0 \implies k - \cos x>0 \implies k>\cos x. \]
Step 3: Maximum value of \( \cos x \)
The maximum value of \( \cos x \) is 1. Therefore: \[ k>1. \]
Step 4: Verify the options
The function is strictly increasing for \( k>1 \), which matches option (A).