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

Answer

The required points are `(0,0)` and `(3,27)`.

Explanation

ATQ, The equation of the curve is `y=x^{3}`.
$$ \therefore \frac{d y}{d x}=3 x^{2} $$
The slope of the tangent at point `(x, y)` is, `\frac{d y}{d x}]_{(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, `y=x^{3}`.
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