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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