Question

Find all points at which the tangents to the curves y = x³ – x + 1 are y = 3x² – 4x + 1 are parallel.

Answer 1

Given curves are

y = x³ –x + 1 and y = 3x² – 4x + 1

Since the tangents are parallel, so their slopes are same

Let the points (α, β) and (γ, δ) on the two given curves where the tangents are parallel

Thus, 3α² – 1 = 6γ – 4

⇒ 3α² – 6γ + 3 = 0

⇒ α² – 2γ + 1 = 0

which is possible for infinite number of ordered pairs

Hence, the number of solutions is infinite.