I am considering the curve traced by the equation $r=a\sin 3\theta$. Specifically as $\theta$ varies from $0$ to $\frac{\pi}{6}$, $r$ varies from $0$ to $a$. How do I conclude that the curve is convex in this domain? If we are dealing with cartesian coordinates for a function $y=f(x)$, the book says that the second derivative being non-negative tells us whether the function is convex or not. But is there such a test in polar coordinates for arbitrary curves? Secondly, if we allow $\theta$ to take all real values, $r$ becomes negative occasionally. Does this not contradict the meaning of r?
Thanks.