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 Jan 20 comment strict local extremum of $f'$ that is neither saddle nor inflection value of $f$ With extremum I mean "strict extremum", then the derivative of constant function would be constant without a strict extremum, so it doesn't match the first condition. Jan 20 asked strict local extremum of $f'$ that is neither saddle nor inflection value of $f$ Dec 22 accepted Relation between points of inflection and saddle points Dec 22 awarded Benefactor Dec 21 comment Relation between points of inflection and saddle points Why did you delete the worked out mean value argument. Was there something wrong with it? Dec 20 awarded Critic Dec 19 comment Relation between points of inflection and saddle points Since the function is differentiable, strictly mono. increasing on the half open intervall implies the same property on the closed intervall. Dec 14 awarded Promoter Dec 13 revised Relation between points of inflection and saddle points added 18 characters in body Dec 11 revised Relation between points of inflection and saddle points added tag Dec 11 revised Relation between points of inflection and saddle points added 16 characters in body Dec 11 comment $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs Thanks, but I noticed that I wasn't precise enough. I want that the property with the second derivative is violated on every neighborhood of $x_0$. See my edit. Dec 11 comment $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs Thanks, but I noticed that I wasn't precise enough. I want that the property with the second derivative is violated on every neighborhood of $x_0$. See my edit. Dec 11 revised $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs added 497 characters in body Dec 11 accepted Does every differentiable function has an infliction point between a local maximum and minimum? Dec 11 asked Relation between points of inflection and saddle points Dec 10 revised $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs added 31 characters in body Dec 9 comment $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs @ClementC. Ideally I want a simple example where one can "write down" a term for $f$ and not only saying that there exists some (maybe very complicated) primitive of your $g$. Dec 9 asked $f'$ changes strict monotonicity but $f''$ isn't of strictly opposite signs Dec 7 revised Chain rule proof from Wikipedia: references? edited body; edited tags; edited title