What Are Local Minimum and Maximum Points in Continuous Functions?

Thread starter sergey_le
Start date Dec 30, 2019
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Minimum Point

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Homework Help Overview

The discussion revolves around identifying local minimum and maximum points in continuous functions, specifically focusing on a scenario where a function has a local minimum at point x1 and a local maximum at point x2, with the condition that f(x1) > f(x2). Participants are exploring the implications of this setup and the necessary conditions for proving the existence of another local minimum.

Discussion Character

Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

Participants discuss the continuity of the function and the implications of local extrema, questioning how to formalize proofs and explore different scenarios. Some suggest sketching functions or using the extreme value theorem to reason about the existence of additional local minima.

Discussion Status

The discussion is active, with participants providing hints and exploring various lines of reasoning. Some have suggested using contradiction or the extreme value theorem to approach the problem, while others are clarifying definitions and conditions necessary for the proof. There is recognition of the complexity involved in formalizing the arguments.

Contextual Notes

Participants note the importance of definitions regarding local minima and maxima, particularly the distinction between strict inequalities and non-strict inequalities in their definitions. There is also mention of the need to ensure that any additional minimum point identified is distinct from the given extrema.