Proving that the tangents to the curve y=x(1-x²) at P and Q are parallel

[Kitimbo]

Proving that the tangents to the curve y=x(1-x²) at P and Q are parallel

Find the coordinates of the point of intersection of the line x-3y=0 with the curve y=x(1-x²).If these points are in order P,O,Q,prove that the tangents to the curve at P and Q are parallel, and that the tangent at O is perpendicular to them.

So I did the first part by solving the two equations simultaneously to get P(-√⅔,-⅓√⅔),O(0,0),Q(√⅔,⅓√⅔)
Help me with proving part

 

[Harry_the_cat]

Find the coordinates of the point of intersection of the line x-3y=0 with the curve y=x(1-x²).If these points are in order P,O,Q,prove that the tangents to the curve at P and Q are parallel, and that the tangent at O is perpendicular to them.

So I did the first part by solving the two equations simultaneously to get P(-√⅔,-⅓√⅔),O(0,0),Q(√⅔,⅓√⅔)
Help me with proving part

Ok so find the gradient at P and Q and show that they are equal (hance the tangents are parallel).

Find the gradient at O and show that it is the negative reciprocal (ie tangent is perpendicular).

(Find the gradients by substituting the x-coordinate of each point into y' where y = x(1-x^2).)