Question

Find the points on the curve y = x³ – 9x² + 15x + 3 at which the tangents are parallel to y-axis.

Answer 1

y = x³ – 9x² + 15x + 3   ....(1)

Differentiate w.r.t. x

`dy/dx` = 3x² – 18x + 15

Since tangents are parallel to the X-axis,

The slope of the tangent is zero.

∴ `dy/dx` = 0,

∴ 3x² – 18x + 15 = 0

∴ x² – 6x + 5 = 0,

∴ (x – 1)(x – 5) = 0

∴ x = 1 or x = 5

If x = 1,

From (1)

y = 1 – 9 + 15 + 3 = 10

If x = 5,

Then y = 125 – 225 + 75 + 3 = – 22

∴ Required points are (1, 10) and (5, – 22)