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}.
Call the second derivative of the polynomial P(x) and for ease of typing, the xcoordinates of the inflection points a,b.
P"(x)=(xa)(xb) = x^2+(ab)x+ab
Integrating twice to get to P(x)
P'(x)=x^3/3+(ab)x^2/2+abx+C
P(x)=x^4/12+(ab)x^3/6+abx^2/2+CX+D
where C and D are constants.
Then we can find the coordinates of the points I1=(a,P(a)) and I2=(b,P(b)) where
P(a)=a^4/12+a^3b/6+Ca+D
P(b)=b^4/12+ab^3/6+Cb+D.
Here's a graph with some sliders:
https://www.desmos.com/calculator/tpjjew5uxx
The value of D obviously doesn't affect the distances in question. The value of C doesn't seem to matter either, since we will only need to consider the xcoordinates of P1 and P2.
Next step: Make some simplifications (b=0,C=0,D=0) and try finding P1 and P2.

Posted by Jer
on 20231124 15:54:03 