I am seeking for a proof of the generalized derivative test to find inflection points, minima and maxima. I am seeking for a proof that I read some time ago but can't find anymore. The thesis was that if the first derivative in a point is zero, and also the second derivative is zero, then you have to continue the derivation until you get an n-th derivative with a non-zero value. If that derivative is of an even order, then you have to check if it is positive or negative and subsequently you deduct that it is a minimum or a maximum. If the order of the derivative is odd then it is an inflection point. The proof used the Taylor formula. I am sure it is a well known subject and that many questions were asked about this, but I can't find a rigorous complete proof, I found only descriptive talks. A link to a PDF is also completely okay. Thanks in advance!:)
Generalized Taylor derivatives test
5
$\begingroup$
calculus
limits
proof-writing
taylor-expansion
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0+1 for an interesting piece of trivia I never knew about! – 2017-01-02
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0I hope that this paper: [Generalized local test for local extrema in single-variable functions](https://faculty.utrgv.edu/eleftherios.gkioulekas/papers/submitted/2nd-deriv-gen.pdf) answers your question. – 2018-02-28
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0That is exactly what I was seeking for. In order to close the question, I should provide an answer in the canonic form, so you could you report the link in a complete answer with some sort of insights or I could do it instead. Thanks for your input! – 2018-03-11