perplexus.info :: Just Math : Inflection Point Line

Inflection Point Line (Posted on 2023-11-24)

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.

No Solution Yet Submitted by Brian Smith

No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)

The finish

Comment 2 of 2 |

Letting b=C=D=0

(I didn't prove the solution is independent of C.)

P(x)=x^4/12-ax^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(x-a)(x^2-ax-a^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 2023-11-24 19:16:15

Please log in:

Login:
Password:
Remember me:

Sign up! \| Forgot password

Search:

Search body:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (12)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info