Question

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.

Answer 1

The equation of the given curve is $y=x^3$ ....(i)

$\frac{dy}{dx}=3x^2$

The slope of the tangent at the points (x,y) is given by ${\left(\frac{dy}{dx}\right)}_{x,y}=3x^2$

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

$\Rightarrow 3x^2=x^3(\therefore y=x^{3}given)$

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

When x=0, then from Eq.(i), we get $y=x^3=0$

$\therefore$ The required point is (0,0).

When x=3, then from Eq.(i), we get $y=3^3=27$

Hence, the required points are (0,0) and (3,27).