Question

Let C be the curve y3 –3xy + 2 = 0. If H is the set of points on the curve C where the tangent is horizontal and V is the set of the point on the curve where the tangent is vertical, then H = ... and V = ...

Answer 1

Given curve is y3 – 3xy + 2 = 0

Since the tangent is horizontal, so dy/dx = 0

⇒ y = 0

which is not possible, since the point y = 0 does not lie on the curve.

Thus, H = ϕ = null set

When the tangent is vertical,so dx/dy = 0

⇒ (y2 – x)/y = 0

⇒ x = y2

when x = y2, then y3 – 3y3 + 2 = 0

⇒ 2y3 = 2

⇒ y3 = 1

⇒ y = 1 when y = 1, then x = 1

Hence, V = {(1, 1)}