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: Calculate the first derivative of \( f(x) \).
A function is monotonically increasing over all real numbers (\( \mathbb{R} \)) if its first derivative is consistently non-negative, meaning:\[f'(x) \geq 0 \quad \forall x \in \mathbb{R}.\]The derivative of \( f(x) \) is computed as:\[f'(x) = \cos x - a.\]Step 2: Determine the condition for \( f'(x) \geq 0 \).
For \( f(x) \) to be increasing on \( \mathbb{R} \), the following inequality must hold:\[\cos x - a \geq 0 \quad \forall x \in \mathbb{R}.\]Given that the range of \( \cos x \) is \( [-1,1] \), its minimum value is \( -1 \) and its maximum value is \( 1 \). Consequently, the condition simplifies to:\[-1 - a \geq 0.\]Which yields:\[a \leq -1.\]Step 3: Final Conclusion.
Therefore, the condition necessary for \( f(x) \) to be increasing on \( \mathbb{R} \) is:\[a \leq -1.\]