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).