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 given by:
\[f'(x) = k - \cos x.\]
Step 2: Determine the condition for an increasing function.
For \( f(x) \) to be strictly increasing, the following condition must hold:
\[f'(x)>0 \implies k - \cos x>0 \implies k>\cos x.\]
Step 3: Identify the maximum value of \( \cos x \).
The maximum value that \( \cos x \) can attain is 1. Consequently:
\[k>1.\]
Step 4: Validate against the options.
The function is strictly increasing when \( k>1 \). This condition aligns with option (A).