Stationary points of an implicit cubic

A curve CC has equation x3+y3−xy=Ax^3+y^3-xy=A, where AA is a non-zero constant.

(a) Show that any stationary points on CC lie on y=3x2y=3x^2. marks

(b) Find, in terms of AA, the xx-coordinates of the stationary points, and hence determine the range of values of AA for which CC has two distinct stationary points. marks

(c) Suppose that CC has two stationary points. Determine the nature of each of the stationary points found in part (b). marks

[2023/EJC/Prelim/II/4]

Total marks: 9

Solution

Choose a part to attempt. Each page keeps the full question available and loads only that part’s working and exploration.

  1. (a) Implicit differentiation →
  2. (b) coordinates Quadratics in a power →
  3. (b) range of A Discriminant →
  4. (c) Second derivative test →