Intervals of increase/decrease and classification of local extrema

First compute $r'(x) = 2\cos(x) + 2\cos(2x)$. Using $\cos(2x) = 2\cos^2(x) - 1$, rewrite this as $r'(x) = 4\cos^2(x) + 2\cos(x) - 2 = 2(2\cos(x) - 1)(\cos(x)+1)$. Setting $r'(x) = 0$ gives $\cos(x) = 1/2$ or $\cos(x) = -1$. On $[0, 2\pi]$ this yields $x = \pi/3, \pi, 5\pi/3$ as critical points.

Testing sample points in each subinterval: on $(0, \pi/3)$ pick $x=\pi/6$: $r'(\pi/6) = 2(2\cos(\pi/6) - 1)(\cos(\pi/6)+1) = 2(\sqrt{3} - 1)(\frac{\sqrt{3}}{2}+1) > 0$ (increasing). On $(\pi/3, \pi)$ pick $x=\pi/2$: $r'(\pi/2) = 2(2\cdot 0 - 1)(0 + 1) = -2 < 0$ (decreasing). On $(\pi, 5\pi/3)$ pick $x=3\pi/2$: $r'(3\pi/2) = 2(2\cdot 0 - 1)(0 + 1) = -2 < 0$ (decreasing). On $(5\pi/3, 2\pi)$ pick $x=11\pi/6$: $r'(11\pi/6) = 2(2\cos(11\pi/6) - 1)(\cos(11\pi/6)+1) = 2(\sqrt{3} - 1)(\frac{\sqrt{3}}{2}+1) > 0$ (increasing).

To find the local extrema values: at $x = \pi/3$, $r(\pi/3) = 2\sin(\pi/3) + \sin(2\pi/3) = 2 \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$. At $x = 5\pi/3$, $r(5\pi/3) = 2\sin(5\pi/3) + \sin(10\pi/3) = 2 \cdot (-\frac{\sqrt{3}}{2}) + (-\frac{\sqrt{3}}{2}) = -\frac{3\sqrt{3}}{2}$. Since $r'$ changes from positive to negative at $x = \pi/3$, this is a local maximum. Since $r'$ changes from negative to positive at $x = 5\pi/3$, this is a local minimum. At $x = \pi$, $r'(\pi) = 0$ but $r'$ remains negative on both $(\pi/3, \pi)$ and $(\pi, 5\pi/3)$, so $r'$ does not change sign at this critical point, making it neither a local maximum nor minimum.

At x = π, the derivative equals zero since cos(π) = -1 makes the factor (cos(x) + 1) equal zero. However, the derivative does not change sign at this point. Since r'(x) = 2(2cos(x) - 1)(cos(x) + 1) is negative immediately before π (where cos(x) > -1) and remains negative immediately after π (where cos(x) > -1), the First Derivative Test cannot classify this as a local extremum. The function is decreasing through x = π with no turning behavior. Therefore this critical point is neither a local maximum nor a local minimum.