Find the points on the curve `y=x^3` at which the slope of the tangent is equal to the y-coordinate of the point.

Explanation

Equation of curve
`y = x^(3)` …(1)
` rArr (dy)/(dx) = 3x^(2)`
Slope of tangent at point `(x, y) = 3x^(2)`
Given ` 3x^(2) = y`
` rArr 3x^(2) = x^(3)` From equation (1)
` rArr x^(2) (3-x)=0`
` rArr x = 0 or x= 3`
put x = 0 in equation (1) we get y = 0
put x = 3 in equation (1) we get ` y=3^(3) = 27`
`:. ` Required point = (0, 0) or (3, 27) .