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

(a)

Implicit differentiation

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

Explore this idea

Hint

  1. Differentiate implicitly.

  2. A stationary point has dydx=0\frac{dy}{dx}=0.

Answer · (a)

Stationary points satisfy y=3x2y=3x^2.

Explore the idea

Open interactive exploration

Flat tangents sit on y = 3x² (shown for A = 1)

Jump to the two stationary points; both satisfy y = 3x².

Position along the curve(0.61536, 1.13601)
Try an example
Tangent gradient
0

Stationary point on the right: x≈0.616x\approx0.616 and y=3x2≈1.14y=3x^2\approx1.14. The tangent is horizontal and the curve peaks here.

The tangent is horizontal and the normal is vertical. Both axes use equal scales.

  • X³ + y³ − xy = 1
  • Tangent
Key idea

A flat tangent forces y=3x2y=3x^2; the curve fixes where on that parabola the point lies.

Ideas to try
Check that each marked point satisfies y=3x2y=3x^2.