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

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The discussion focuses on identifying specific characteristics of functions and their derivatives, particularly points labeled A, B, C, D, and E. Points B and E are confirmed to have non-zero first and second derivatives with the same sign, while points A, C, and D are debated for having at least two of the functions f, f', and f'' equal to zero. The participants analyze the implications of slopes and inflection points, concluding that point D may not belong in the selection due to its characteristics regarding the axes.

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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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