Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
The equation of the given curve is y = x3.
`:. dy/dx = 3x^2`
The slope of the tangent at the point (x, y) is given by,
When the slope of the tangent is equal to the y-coordinate of the point, then y = 3x2.
Also, we have y = x3.
∴3x2 = x3
⇒ x2 (x − 3) = 0
⇒ x = 0, x = 3
When x = 0, then y = 0 and when x = 3, then y = 3(3)2 = 27.
Hence, the required points are (0, 0) and (3, 27).