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

If this part uses an earlier result, revisit (b) range of A Discriminant or open all working together.

(c)

Second derivative test

Determine the nature of each stationary point.

Explore this idea

Hint

  1. Differentiate (3y2−x)dydx=y−3x2(3y^2-x)\frac{dy}{dx}=y-3x^2 again.

  2. Put dydx=0\frac{dy}{dx}=0 and y=3x2y=3x^2.

  3. Use the sign of d2ydx2\frac{d^2y}{dx^2}.

Answer · (c)

Plus root: maximum. Minus root: minimum.

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Open interactive exploration

A maximum and a minimum on the same curve

Jump to each stationary point and look at which way the curve bends.

Position along the curve(-0.54169, 0.88027)
Try an example
Tangent gradient
0

Minimum: d2ydx2=61+27A>0\frac{d^2y}{dx^2}=\frac6{\sqrt{1+27A}}>0. The curve bends up and the tangent lies below it.

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

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

Positive second derivative: minimum. Negative: maximum. The sign comes entirely from the cube root sign.

Ideas to try
On which side of the tangent does the curve lie at each point?