For which real numbers c is there a straight line that intersects the curve

y = x⁴ + 9x³ + cx² + 9x + 4

in four distinct points?

I believe the only way to achieve the requested result is for the curve to have three distinct points for maxima/minima.  In other words, if the first derivative has any repeated roots, then the shape of the curve will not allow for 4 distinct intersects with any line.

Posted by Kenny M on 2024-09-04 04:45:19