In my textbook there is a confusing statement.
If $f'''(ξ)=0 $ and $f''(ξ)\ne0$ then $ξ$ is inflection point. However this confuses me as it is contrary to book example and this.
Also in class notes it is the same as far as I can see but shouldnt derivatives be in different order? Second first and then third.
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$\begingroup$ I think your textbook is just wrong, or at best is indulging in a deviant definition of inflection point. The Wikipedia article on the subject (not the one you refer to) gives the conventional view. $\endgroup$– ThumbnailCommented Nov 28, 2014 at 11:57
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$\begingroup$ Well what has confused me in first place was its example where it finds f''(2)=0 for a point so they check the third f'''(2)=6 and since its !=0 it states that it is a inflection point . But this doesnt have any relation with the statement... $\endgroup$– GorillaApeCommented Nov 28, 2014 at 12:05
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it must be $$f''(\xi)=0$$ and $$f'''(\xi)\ne 0$$ or higher derivatives of odd degree.
answered Nov 28, 2014 at 11:38
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$\begingroup$ So are you 100% sure that book is wrong? $\endgroup$ Commented Nov 28, 2014 at 11:51
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$\begingroup$ Taking the example of $f(x) = x^2$ at the origin, which is not an inflection point, but has $f''(0) \ne 0$ and $f'''(0) = 0$, I'm 100% certain that the book is wrong. $\endgroup$ Commented Nov 28, 2014 at 13:12