Question

Find the values of \( a \) for which \( f(x) = \sqrt{3} \sin x - \cos x - 2ax + b \) is decreasing on \( \mathbb{R} \).

Hint:

To determine when a function is increasing or decreasing, compute its derivative and analyze when it is positive or negative.

Answer 1

To determine the values of \( a \) for which \( f(x) \) is decreasing, we compute its derivative \( f'(x) \). The derivative is given by: \[ f'(x) = \sqrt{3} \cos x + \sin x - 2a \] For \( f(x) \) to be decreasing over all real numbers \( \mathbb{R} \), the condition \( f'(x) \leq 0 \) must hold for all \( x \). The term \( \sqrt{3} \cos x + \sin x \) is bounded as it is a sum of sinusoidal functions. Its maximum value is \( \sqrt{(\sqrt{3})^2 + 1^2} = 2 \). Therefore, we must satisfy the inequality: \[ 2 - 2a \leq 0 \] Solving this inequality for \( a \) yields: \[ a \leq 0 \] Consequently, \( f(x) \) is decreasing for all \( x \) when \( a \leq 0 \).