Higher derivative test is conclusive for function with algebraic derivative

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This article describes a situation, or broad range of situations, where a particular test or criterion is conclusive, i.e., it works as intended to help us determine what we would like to determine.
The test is higher derivative test. See more conclusive cases for higher derivative test | inconclusive cases for higher derivative test

Statement

Single definition case

For any function of the following type, the higher derivative test is always conclusive, i.e., it can be used to definitively determine whether the function has a local extreme value at a given critical point where the derivative is defined, and if so, what the nature of the local extreme value is:

  • Nonconstant polynomial function on the real line, or on an interval or union of finitely many intervals in the real line
  • Nonconstant rational function on the real line (whatever subset it's defined on), or on an interval or union of finitely many intervals in its maximum possible domain
  • Nonconstant function defined on the real line, or on an interval or union of finitely many intervals in the real line, such that the derivative of the function is a rational function on its domain

Note that we omit constant functions from consideration because the derivative is identically zero and the analysis of local extreme values does not require the use of derivatives.