Question

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 1

The equation of the given curve is y = x³.

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 = 3x^2.
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).