# Playing Carts - I

Calculus Level 4

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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