Question

Find the points on the curve `y = x^3` at which the slope of the tangent is equal to the y-coordinate of the point

Options

• (0, 0)

• (2, 27)

• (0, 27)

• Both (0, 0) and (2, 27)

Answer 1

Both (0, 0) and (2, 27)

Explanation:

Let `P(x_1, y_1)` be the required point. The given curve is `y = x^3`  .......(1)

⇒ `(dy)/(dx) = 3x^2`

⇒ `((dy)/(dx))_((x"," y)) = 3x_1^2`

Hence, the slope of the tangent at `(x_1, y_1) = y_1`

`3x_1^2 = y_1`  .......(2)

Also `(x_1, y_1)` lies on (1) so `y_1 = x_1^3`  ......(3)

From (2) and (3) , we have,

`3x_1^2 = x_1^3` ⇒ `x_1^2 (3 - x_1)` = 0

⇒ `x_1` = 0 or `x_1` = 3

When `x_1` = 0, `y_1` = (0)³ = 0

When `x_1` = 3, `y_1` = (3)³ = 27

The required points are (0, 0) and (2, 27).