Mathematics

This problem asks you to find where the curve has horizontal or vertical tangents (critical points), then use the second derivative to classify them. The key insight is that you'll need implicit differentiation to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, and critical points occur where $\frac{dy}{dx} = 0$ or is undefined. When $\frac{d^2y}{dx^2} = 0$ at a critical point, the second-derivative test fails because you can't determine concavity—you'll need to analyze the curve's behavior directly near that point to decide if it's a local extremum or saddle point.

This problem asks you to find where the curve has horizontal or vertical tangents (critical points), then use the second derivative to classify them. The key insight is that you'll need implicit differentiation to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$, and critical points occur where $\frac{dy}{dx} = 0$ or is undefined. When $\frac{d^2y}{dx^2} = 0$ at a critical point, the second-derivative test fails because you can't determine concavity—you'll need to analyze the curve's behavior directly near that point to decide if it's a local extremum or saddle point.