Why is the statement "An inflection point can not be a local extreme point?" wrong? Isn't a local extreme point either a max or a min only? What is wrong here?
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The phrasing of the statement is awkward in itself, and there is an overall lack of context in the question. – Arnaud Mortier Jul 12 '18 at 11:39
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Think of the problem $\min x^3$ s.t. $x\ge 0$ Here $x = 0$ is an inflection point but is also the solution point. – Cesareo Jul 12 '18 at 11:39
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@Cesareo: Some authors require local extrema to be two-sided. If a local extremum can be one-sided, and an inflection point can also be one-sided, then your counterexample is perfect. – Adrian Keister Jul 12 '18 at 16:10
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@AdrianKeister What do you mean by a "one-sided inflection point"? – user539887 Jul 12 '18 at 20:43
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@user539887: I don't think they exist, which was sort of my point. The concavity has to change at a PoI, which means there has to be function to the left and to the right, making Cesareo's counterexample not a counterexample. – Adrian Keister Jul 12 '18 at 20:44
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@Adrian Keister: The precise meaning of "inflection point" varies in the literature (see the papers I cited in this comment, for example), and so we probably need to know exactly what definition of "inflection point" the OP is working with. See also this 24 December 2005 ap-calculus post (and this followup) archived at Math Forum. – Dave L. Renfro Jul 12 '18 at 20:59