MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Let the curve x2y+y3=6x be given.
(i) Derive an expression for dxdy in its simplest form.
(ii) Show that every point on the curve satisfying xy=3 has a horizontal tangent, and find the exact coordinates of all such points.
(iii) Investigate whether the curve has any vertical tangents and justify your conclusion algebraically.
(iv) For each point obtained in part (ii), write the equation of the tangent line and of the normal line.