Box Up The Cubic

Calculus Level 5

For a certain value \(p\), there can be only one box that is tangent on all 4 sides to this cubic \({ x }^{ 3 }-px\). Find the ratio of its sides. (The ratio is a number greater than 1.)

The box is a rectangle, possibly tilted. The cubic curve may pass through the box, but this cubic curve must nevertheless be tangential to all four sides of the box.

Give your answer to 3 decimal places.

Note: The above image is an example of a box tangent on all 4 sides to a cubic. However, this box is not the only one possible for this cubic.


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