Question

Find the values of \( a \) for which \( f(x) = \sin x - ax + b \) is increasing on \( \mathbb{R} \).

Hint:

A function is increasing when its first derivative is non-negative. Consider the maximum and minimum values of trigonometric functions when solving inequalities.

Answer 1

Step 1: Find the first derivative of \( f(x) \).
A function is increasing on \( \mathbb{R} \) if its first derivative is always non-negative, i.e., \[ f'(x) \geq 0 \quad \forall x \in \mathbb{R}. \] Differentiating \( f(x) \): \[ f'(x) = \cos x - a. \] Step 2: Find the condition for \( f'(x) \geq 0 \).
For \( f(x) \) to be increasing on \( \mathbb{R} \), we must have: \[ \cos x - a \geq 0 \quad \forall x \in \mathbb{R}. \] Since \( \cos x \) oscillates in the range \( [-1,1] \), the minimum value of \( \cos x \) is \( -1 \), and the maximum value is \( 1 \). Therefore, the condition becomes: \[ -1 - a \geq 0. \] \[ a \leq -1. \] Step 3: Conclusion.
Thus, for \( f(x) \) to be increasing on \( \mathbb{R} \), the required condition is: \[ a \leq -1. \]