Question

Find the points on the curve ${\displaystyle y=x^3}$ at which the slope of the tangent is equal to the y-coordinate of the point

विकल्प

• (0, 0)

• (2, 27)

• (0, 27)

• Both (0, 0) and (2, 27)

Answer 1

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

Explanation:

Let ${\displaystyle P(x_1,y_1)}$ be the required point. The given curve is ${\displaystyle y=x^3}$  .......(1)

⇒ ${\displaystyle \frac{dy}{dx}=3x^2}$

⇒ ${\displaystyle {\left(\frac{dy}{dx}\right)}_{\left(x\text{,}y\right)}=3x_1^2}$

Hence, the slope of the tangent at ${\displaystyle (x_1,y_1)=y_1}$

${\displaystyle 3x_1^2=y_1}$  .......(2)

Also ${\displaystyle (x_1,y_1)}$ lies on (1) so ${\displaystyle y_1=x_1^3}$  ......(3)

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

${\displaystyle 3x_1^2=x_1^3}$ ⇒ ${\displaystyle x_1^2(3-x_1)}$ = 0

⇒ ${\displaystyle x_1}$ = 0 or ${\displaystyle x_1}$ = 3

When ${\displaystyle x_1}$ = 0, ${\displaystyle y_1}$ = (0)³ = 0

When ${\displaystyle x_1}$ = 3, ${\displaystyle y_1}$ = (3)³ = 27

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