A quartic polynomial is given to have two distinct inflection points, call those points I_{1} and I_{2}. A line is drawn through these inflection points and intersects the quartic at two other points, call those points P_{1} and P_{2}.
The four points will lay along the line in the order P_{1}, I_{1}, I_{2}, P_{2}.
Show that segments P_{1}I_{1} and P_{2}I_{2} are congruent.
Find the ratio of segment P_{1}I_{1} to I_{1}I_{2}.
Letting b=C=D=0
(I didn't prove the solution is independent of C.)
P(x)=x^4/12ax^3/6
P(a)=a^4/12
So the line passes through (0,0) and (a,P(a)) and has eq. y=a^3x/12
Find the intertsection of this line with y=P(x). This involves solving a degree 4 equation but since we know it has roots at 0 and a so it can be factored as
x(xa)(x^2axa^2)
The final factor will give P1 and P2
Solving this quadratic gives
x=a(1+/sqrt5)/2
The positive root is the golden ratio = phi
P1I1=P2I2=a/phi
P1I2/I1I2=phi

Posted by Jer
on 20231124 19:16:15 