SAT1000 - P966

For $f(x) = x - \frac 13 \sin 2x + a \sin x$ to be monotonic increasing for all real $x$ , $f'(x)$ must be non-negative or

$\begin{aligned} f'(x) & \ge 0 \\ 1 - \frac 23 \cos 2x + a \cos x & \ge 0 \\ 1 - \frac 43 \cos^2 x + \frac 23 + a \cos x & \ge 0 \\ 4 \cos^2 x - 3a \cos x - 5 & \le 0 \\ 4 \blue{\left(\cos x - \frac {3a}8 \right)^2} - \frac {9a^2}{16} - 5 & \le 0 & \small \blue{\max \left(\cos x - \frac {3a}8 \right)^2 = \left| 1 + \frac {3|a|}8 \right|^2} \\ 4 \blue{\left|1+ \frac {3|a|}8 \right|^2} - \frac {9a^2}{16} - 5 & \le 0 & \small \blue{\text{when }\cos x = \begin{cases} -1 & \text{for }a > 0 \\ 1 & \text{for }a < 0 \end{cases}} \\ \frac {9a^2}{16} + 3|a| + 4 - \frac {9a^2}{16} & \le 5 \\ 3 |a| & \le 1 \\ |a| & \le \frac 13 \\ \implies a & \in \boxed{\left[-\frac 13,\ \frac 13 \right]} \end{aligned}$