Find the points on the curve

The equation of the given curve is $y=x^{3}$.

$\therefore \frac{d y}{d x}=3 x^{2}$

The slope of the tangent at the point (x, y) is given by,

$\left.\frac{d y}{d x}\right]_{(x, y)}=3 x^{2}$

When the slope of the tangent is equal to the $y$-coordinate of the point, then $y=3 x^{2}$.

Also, we have $y=x^{3}$.

$\therefore 3 x^{2}=x^{3}$

$\Rightarrow x^{2}(x-3)=0$

$\Rightarrow x=0, x=3$

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).