If a cubic equation has real zeroes +p, p and 0, and its two horizontal tangent points and two nonzero xintercepts can be joined together to form a rectangle, then find the ratio of the rectangleâ€™s larger side to its shorter side.
If the polynomial is f(x)=x(xp)(x+p) then the horizontal tangent points are where the derivative equals zero. That is
f'(x)=3x^2p^0=0
x=+/ p/sqrt(3)
The point in the 4th quadrant is (p/sqrt(3), 2p^3/3sqrt(3))
The quadrilateral described is a parallelogram because of its rotational symmetry. One way to make it a rectangle is to make the four corners equidistant from the origin:
Out of time here...
The solution is an interesting fourth root:
p=(9/2)^(1/4)
I didn't finish finding the ratio.

Posted by Jer
on 20190307 15:57:25 