Question

The points at which the tangents to the curve y = x³ – 12x + 18 are parallel to x-axis are:

A. (2, –2), (–2, –34)
B. (2, 34), (–2, 0)
C. (0, 34), (–2, 0)
D. (2, 2), (–2, 34)

Answer 1

Given the equation of the curve is

y = x³ – 12x + 18

Differentiating on both sides with respect to x, we get

We know derivative of a constant is 0, so above equation becomes

So equating equation (i) to 0, we get

3x²-12=0

⇒ 3x²=12

⇒ x²=4

⇒ x=±2

When x=2, the given equation of curve becomes,

y = x³ – 12x + 18

⇒ y = (2)³ – 12(2) + 18

⇒ y = 8– 24 + 18

⇒ y = 2

When x=-2, the given equation of curve becomes,

y = x³ – 12x + 18

⇒ y = (-2)³ – 12(-2) + 18

⇒ y = -8+24 + 18

⇒ y = 34

Hence the points at which the tangents to the curve y = x³ – 12x + 18 are parallel to x-axis are (2, 2) and (-2, 34).

So the correct option is option D.