Box Up The Cubic
For a certain value ,
there can be only one box that is tangent on all 4 sides to this cubic . 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.
The answer is 4.000.
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1 solution
We seek to find a box where two of the tangent points are at opposite corners. The equation for the line tangent to x 3 − p x at x = a is ( 3 a 2 − p ) x − 2 a 2 , which intersects the cubic at x = − 2 a . The equation for the line tangent to x 3 − p x at x = − 2 a is ( 1 2 a 2 − p ) x + 1 6 a 2 . For a box to be formed, then, the slopes of the tangent lines must be orthogonal, which is true if ( 3 a 2 − p ) = − ( 1 2 a 2 − p ) − 1 , which is true if a = ± 2 6 1 5 p ± 9 p 2 − 1 6 . An unique solution exists only if 9 p 2 − 1 6 = 0 , or p = 3 4 . From this,the ratio of the sides of the box can eventually be worked out to be exactly 4 . For p < 3 4 , no box is possible where all of the tangent points are confined to its 4 sides. For p > 3 4 , more than one such box is possible.