Point P is a point of inflection?

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Discussion Overview

The discussion revolves around the conditions under which a point P can be classified as a point of inflection in the context of calculus. Participants explore the implications of the first, second, and third derivatives at point P.

Discussion Character

  • Debate/contested

Main Points Raised

  • Hoot questions whether the conditions f'(x) = 0, f''(x) = 0, and f'''(x) \neq 0 definitively indicate that point P is a point of inflection.
  • Another participant suggests that the definition of "point of inflection" can vary, implying that the classification may depend on the context or specific definitions used.
  • A counterexample is provided involving the function x^5 at x=0, which challenges the reliability of the test for identifying points of inflection.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of a point of inflection, indicating that multiple competing views remain regarding the criteria for classification.

Contextual Notes

The discussion highlights the ambiguity in the definition of a point of inflection and the potential for differing interpretations based on specific mathematical contexts.

Hootenanny
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I just want to check something I'm not sure on. If at point P; [itex]f'(x) = 0[/itex] and [itex]f''(x) = 0[/itex] and [itex]f'''(x) \neq 0[/itex] can we definatly say that point P is a point of inflection?

Regards,
~Hoot
 
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"bump" o:)
 
That depends how you define "point of inflection", some times that's just how it is defined.

Easy way to test, if you consider x5 to be a point of inflection at x=0 then your test fails.

However if you don't consider it to be a point of inflection then your test is fine.
 
Last edited:
Thank you Zurtex, extremely helpful.

Regards,
~Hoot
 

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