Mathematics

A rose curve traces out petals as $\theta$ varies, and an inflection point is where the curve's concavity switches. The key is to find where the second derivative $\frac{d^2y}{dx^2}$ changes sign—not just where it equals zero. You'll need to convert to Cartesian coordinates using $x = r\cos\theta$ and $y = r\sin\theta$, then compute $\frac{d^2y}{dx^2}$ using the chain rule carefully. The inflection points won't necessarily align with where $r$ or $\frac{dr}{d\theta}$ equal zero; you're looking for genuine changes in the curve's bending direction.

A rose curve traces out petals as $\theta$ varies, and an inflection point is where the curve's concavity switches. The key is to find where the second derivative $\frac{d^2y}{dx^2}$ changes sign—not just where it equals zero. You'll need to convert to Cartesian coordinates using $x = r\cos\theta$ and $y = r\sin\theta$, then compute $\frac{d^2y}{dx^2}$ using the chain rule carefully. The inflection points won't necessarily align with where $r$ or $\frac{dr}{d\theta}$ equal zero; you're looking for genuine changes in the curve's bending direction.