Answer
$(0,\frac{\pi}{2}),(\frac{\pi}{2},\pi),(\pi,\frac{3\pi}{2}),(\frac{3\pi}{2},2\pi)$
Work Step by Step
$r(\theta)= 2\cos^3\theta \textbf i + 3\sin^3\theta \textbf j\\
r'(\theta)= 6\cos^2\theta \sin \theta \textbf i + 9\sin^2\theta \cos \theta \textbf j $
So, $r'(\theta)= 0 \textbf i + 0 \textbf j $ for $\theta = 0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi$.
Therefore, the curvevis smooth in the intervals
$(0,\frac{\pi}{2}),(\frac{\pi}{2},\pi),(\pi,\frac{3\pi}{2}),(\frac{3\pi}{2},2\pi)$
