What is the greatest value of $f(x) = 2\sin x + \sin 2x$ on $[0, 2\pi]$?

Explanation

1. 01

   $f'(x) = 2\cos x + 2\cos 2x = 0$ gives $2\cos^{2} x + \cos x - 1 = 0$: $\cos x = \frac{1}{2}$ or $-1$, so $x = \frac{\pi}{3}, \pi, \frac{5\pi}{3}$.

2. 02

   $f\left(\frac{\pi}{3}\right) = \sqrt{3} + \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$, $f(\pi) = 0$, $f\left(\frac{5\pi}{3}\right) = -\frac{3\sqrt{3}}{2}$, and 0 at both ends.