If a cubic equation has real zeroes +p, -p and 0, and its two horizontal tangent points and two non-zero x-intercepts can be joined together to form a rectangle, then find the ratio of the rectangle’s larger side to its shorter side.
If the polynomial is f(x)=x(x-p)(x+p) then the horizontal tangent points are where the derivative equals zero. That is
f'(x)=3x^2-p^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.
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Posted by Jer
on 2019-03-07 15:57:25 |
Most of solution