Which Points Have Specific Characteristics for f', f'', and f'''?

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Discussion Overview

The discussion revolves around identifying specific characteristics of the function \( f \) and its derivatives \( f' \) and \( f'' \) at various labeled points. Participants are analyzing the conditions under which these derivatives are non-zero, equal to zero, or share the same sign, focusing on points A, B, C, D, and E.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that points B and E have \( f' \) and \( f'' \) non-zero and the same sign.
  • Others suggest that points A, C, and D have at least two of \( f, f', f'' \) equal to zero, but there is uncertainty about the inclusion of point D.
  • One participant questions the inclusion of point D, noting that it does not have zeros on the x or y-axis but has a slope of zero.
  • Another participant argues for keeping point D, stating that the slope appears to be zero and that the curvature is changing there.
  • There is a suggestion to reconsider point C, as it has \( f = 0 \) and \( f' = 0 \), but concerns are raised about the absence of an inflection point at \( f'' \).
  • One participant concludes that point C does not belong, arguing that \( 0 < f \) and \( f'' < 0 \) at that point.

Areas of Agreement / Disagreement

Participants express differing views on the inclusion of points C and D, indicating that multiple competing perspectives remain regarding the characteristics of these points.

Contextual Notes

There are unresolved questions about the definitions of "zeros" in this context and the implications of slopes and curvature at the specified points.

karush
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Which of the points labeled by a letter have

(a) $f'$ and $f''$ non-zero and the same sign B, E

(b) At least two of $f, f', f''$ equal to zero A, C, D

not sure if these selections were correct and was ? about D in the slope appears to be zero

f' is about slope f'' is about inflection pts and increasing and decreasing I presume

thanks ahead.
 
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I agree with your answer for a), but I would remove one of the points from b). I don't want to say which, as I want you to re-examine on your own.
 
well, i would remove D since has neither zero on x or y-axis but my ? is it does have a m=0

not absolute sure what is meant by zero's here
 
I would keep D, as the slope appears to be zero there, and it appears that curvature is changing there also.
 
ok how about C since at f it has x=0 at f' it has m=0 but f'' there is no inflection pt.
or is it ...
 
Yes, I think C does not belong, since at that point it appears to me that 0 < f and f'' < 0.
 

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