Mathematics

When $\frac{dy}{dx}=0$, the curve has a horizontal tangent—these are your candidate extrema. Use implicit differentiation to find $\frac{dy}{dx}$ in terms of $x$ and $y$, then set it equal to zero to find which points satisfy this condition. Once you've located these points, apply the second derivative test using $\frac{d^2y}{dx^2}$ to determine whether each represents a local max, local min, or saddle point.

When $\frac{dy}{dx}=0$, the curve has a horizontal tangent—these are your candidate extrema. Use implicit differentiation to find $\frac{dy}{dx}$ in terms of $x$ and $y$, then set it equal to zero to find which points satisfy this condition. Once you've located these points, apply the second derivative test using $\frac{d^2y}{dx^2}$ to determine whether each represents a local max, local min, or saddle point.