Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point

Solution:

For a curve y = f(x) containing the point (x₁,y₁) the equation of the tangent line to the curve at (x₁,y₁) is given by 

y − y₁ = f′(x₁) (x − x₁)

The equation of the given curve is y = x³

Therefore,

dy/dx= 3x²

When the slope of the tangent is equal to the y-coordinate of the point,

then according to the question,

y = 3x²

Also, we have y = x³

Therefore,

3x² = x³

⇒ x² (x - 3) = 0

⇒ x = 0, x = 3

When, x = 0,

⇒ y = 3 and x = 3,

⇒ y = 3(3)² = 27

Thus, the points are (0, 0) and (3, 27)

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.3 Question 17

Find the points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point

Summary:

The points on the curve y = x³ at which the slope of the tangent is equal to the y-coordinate of the point are (0, 0) and (3, 27)