The folium-like curve x3+y3−9xy=0 is shown implicitly in the xy–plane.
(1) Derive a simplified expression for dxdy at an arbitrary point (x,y) on the curve.
(2) Determine all points on the curve where the tangent line is horizontal. Present exact coordinates.
(3) For each point found in part (b), compute dx2d2y and decide whether the curve is concave up or concave down with respect to the x–axis at that point. Show analytical justification.
(4) Treating y as a differentiable function of x, classify each point from part (b) as a local maximum, local minimum, or neither for y. Support your conclusions with appropriate reasoning.