What is the plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques called?

The story behind the answer

The Folium of Descartes is the plane curve René Descartes proposed to challenge Pierre de Fermat’s methods for finding extrema and tangents. Defined by x^3 + y^3 = 3axy (a > 0), it has a distinctive leaf-shaped loop, making it an ideal test case for early calculus techniques.

Context: Descartes and Fermat were rivals who helped launch analytic geometry and methods that foreshadowed calculus. The folium exposed limitations in existing extremum/tangent rules and spurred more general approaches. Memory tip: “Folium” means “leaf” in Latin—picture the curve’s loop as a leaf—and “of Descartes” links it to the founder of Cartesian geometry.