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 find the values of \( a \) for which \( f(x) \) is decreasing, we first need to compute the derivative \( f'(x) \). The derivative of \( f(x) \) is: \[ f'(x) = \sqrt{3} \cos x + \sin x - 2a \] For \( f(x) \) to be decreasing on \( \mathbb{R} \), we need \( f'(x) \leq 0 \) for all \( x \). The expression \( \sqrt{3} \cos x + \sin x \) is bounded, since it is the sum of sinusoidal functions. The maximum value occurs when: \[ \sqrt{3} \cos x + \sin x = \sqrt{(\sqrt{3})^2 + 1^2} = 2 \] Thus, we have the inequality: \[ 2 - 2a \leq 0 \] Solving for \( a \), we get: \[ a \leq 0 \] Therefore, the values of \( a \) for which \( f(x) \) is decreasing are \( a \leq 0 \).