A quartic polynomial is given to have two distinct inflection points, call those points I1 and I2. A line is drawn through these inflection points and intersects the quartic at two other points, call those points P1 and P2.
The four points will lay along the line in the order P1, I1, I2, P2.
Show that segments P1I1 and P2I2 are congruent.
Find the ratio of segment P1I1 to I1I2.
Call the second derivative of the polynomial P(x) and for ease of typing, the x-coordinates of the inflection points a,b.
P"(x)=(x-a)(x-b) = x^2+(-a-b)x+ab
Integrating twice to get to P(x)
P'(x)=x^3/3+(-a-b)x^2/2+abx+C
P(x)=x^4/12+(-a-b)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 x-coordinates of P1 and P2.
Next step: Make some simplifications (b=0,C=0,D=0) and try finding P1 and P2.
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Posted by Jer
on 2023-11-24 15:54:03 |
A start?
