Playing Carts - II

Algebra Level pending

Consider a Cartesian plane where the functions \( f(x) = x^3, \; f'(x), \; f''(x) \) are all plotted.

Let \(\triangle{\alpha}\) be the triangle with vertexes on all lattice points for which one of the functions intersects another.

Let \(\triangle{\beta} \) be the triangle with vertexes on the first quadrant points for which one of the functions intersects another.

(1) The distance between \( \alpha \)'s midpoint to the origin of the Cartesian plane is greater than the one between \( \beta \)'s midpoint to the origin.

(2) \( \alpha \) contains the greatest side length compared to \( \beta\).

(4) Both triangles are obtuse.

(8) \( \alpha \)'s circumradius is greater than \( \beta \)'s.

(16) The parabola passing through all \( \beta\)'s vertexes has real roots.

Find the sum of the numbers in all correct given statements

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