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: {Calculate the derivative}
The derivative of \( f(x) \) is:\[f'(x) = k - \cos x.\]Step 2: {Determine the condition for an increasing function}
For \( f(x) \) to be strictly increasing, the derivative must be positive:\[f'(x)>0 \implies k - \cos x>0 \implies k>\cos x.\]Step 3: {Identify the maximum value of \( \cos x \)}
The maximum value of the cosine function is 1. Consequently:\[k>1.\]Step 4: {Validate the result against options}
The function is strictly increasing when \( k>1 \), which corresponds to option (A).