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

Evaluate the **approximate** numerical value of the sum of \( \alpha \)'s area and perimeter plus \( \beta \)'s area and perimeter.

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