Is there a way to find "polynomial rational functions" having point of inflections, in their graphs?

kolutastmr 2022-07-03 Answered
Is there a way to find "polynomial rational functions" having point of inflections, in their graphs?
For example if the derivative of f(x) is
f ( x ) = ( x 1 ) 2 ( x 3 ) ( x 2 )
then f(x) has a point of inflection at x = 1.
But f(x) is not a polynomial rational !
Is there a way to determine which antiderivatives are the ratio of to polynomials?
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Answers (1)

vrtuljakc6
Answered 2022-07-04 Author has 16 answers
Step 1
x 0   is an inflection point of f f ( x 0 ) = 0   and   f ( x 0 ϵ ) f ( x 0 + ϵ ) < 0
where x 0 + ϵ and x 0 ϵ are in the neighborhood of x 0 .
Step 2
However, you should keep in mind that a denominator that is not constant in a function, in general has an integral with logarithms or with another transcendent function such as trigonometers, for example 1 1 + x 2 d x = arctan ( x ) + C.
I thing the answer to your question is negative.
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