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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Is there a way to find &quot;polynomial rational functions&quot; 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?

vrtuljakc6 · Accepted Answer

Step 1      x    0       is an inflection point of f    ⟺        f    ″    (      x    0    )  =  0     and         f    ″    (      x    0    −  ϵ  )      f    ″    (      x    0    +  ϵ  )  &amp;lt;  0where       x    0    +  ϵ and       x    0    −  ϵ are in the neighborhood of       x    0  . Step 2However, 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.

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

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