Critical point of a piecewise function

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Discussion Overview

The discussion revolves around the concept of critical points in the context of a piecewise function defined on a closed interval. Participants explore the definition of critical points, particularly in relation to differentiability and the behavior of the function at specific points.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether a point where the function is not differentiable can be considered a critical point.
  • Several participants seek clarification on the definition of a critical point, with one providing a definition from Wikipedia that includes points where the function is not differentiable or where the derivative is zero.
  • Another participant notes that different definitions of critical points exist, suggesting that the answer may depend on the specific definition being used.
  • There is a discussion about the relevance of critical points in finding minima or maxima, with a reference to evaluating points in the context of a "V" shaped function.
  • One participant argues that since the function is not defined at x=0, it cannot be considered a critical point.

Areas of Agreement / Disagreement

Participants express differing views on the definition of critical points and whether non-differentiable points qualify. There is no consensus on which definition should be applied in this context.

Contextual Notes

Participants highlight the ambiguity in definitions and the implications for the problem at hand, indicating that the discussion is influenced by varying interpretations of critical points.

mohammed El-Kady
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TL;DR
critical point to piece wise function
If the function is not differentiable at point. Can we consider this point is critical point to the function?
f(x) = (x-3)^2 when x>0
= (x+3)^2 when x<0
he asked for critical points in the closed interval -2, 2
 
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Please define critical point.
 
Math_QED said:
Please define critical point.
you don't know it?
 
" When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero." definition from Wikipedia Looks like the answer is yes. (derivative discontinuous)
 
mohammed El-Kady said:
you don't know it?

Different people use different definitions... The answer to your question depends on the definition you are using.
 
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mathman said:
" When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero." definition from Wikipedia Looks like the answer is yes. (derivative discontinuous)
thank you
 
Math_QED said:
Different people use different definitions... The answer to your question depends on the definition you are using.
I don't know which definition the problem ask for. I know the definition of the derivative, but someone solved it with the differentiability , so i need to be sure from it.
 
critical point vis-a-vis finding minima/maxima? Yes, you have to evaluate those points. Think of a "V" shaped function. How else would you find the minimum.
 
I don't think you meant to do this, but you have not defined the function at x=0, so it is not a critical point of the function.
 

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