Find the points on the curve y = x3 at which the slope of the tangent

1.	Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
Answer» y = x³ \(\frac{dy}{dx}\)= 3x², Given that \(\frac{dy}{dx}\) = y = x³ ∴ 3x² = x³ ⇒ x² (3 – x) = 0 ⇒ x = 0 or x = 3 when x = 3, y = 33 = 27, when x = 0, y = 0 ∴ The required points are (0, 0), (3, 27).

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

Answer»

y = x³

\(\frac{dy}{dx}\)= 3x²,

Given that \(\frac{dy}{dx}\) = y = x³

∴ 3x² = x³ ⇒ x² (3 – x) = 0

⇒ x = 0 or x = 3

when x = 3, y = 33 = 27,

when x = 0, y = 0

∴ The required points are (0, 0), (3, 27).