Help

Mar 2019
4
3
Italy
Hey guys, this is a problem I have come to deal with in math class. It asks to show that the locus of the inflection point of f(x) (the first function, k is a real, non-null number) "comes from" (literally "comes from", this is maybe the reason why it isn't clear to me) f(x) (the second function).
I have tried double differentiating the first function and I find that the locus generated by the inflection point is a hyperbola. The second derivative of the second function is also a hyperbola. The question is how should I put together my demonstration? Is it even right? I am quite puzzled as I don't understand what I should be doing.
Thanks in advance to anyone who will take the time to help!
(I am still in high school, btw, but we do calculus, that's why I put this in the calculus section even though I'm not at University.)


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Jun 2013
1,074
561
Lebanon
Inflection point:

\(\displaystyle \left( x= \frac{1-k}{3 k} ,y= \frac{-2+15 k+12 k^2+2 k^3}{27 k^2}\right)\)

Eliminate $k$