Does the Second Derivative Test Fail for x^3 at x=0?

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Discussion Overview

The discussion revolves around the application of the second derivative test to the function f(x) = x³ at the critical point x = 0. Participants explore whether the test fails in this case and what implications that has for identifying the nature of the critical point.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that since f'(0) = 0 and f''(0) is positive, it should indicate a local maximum, but this is contradicted by the behavior of the function.
  • Another participant claims that f''(0) is not positive, stating it equals zero, which leads to the conclusion that the second derivative test fails.
  • A later reply suggests that the second derivative test does not fail but rather indicates the possibility of an inflection point.
  • Another participant counters that the second derivative test fails to distinguish between maxima, minima, or inflection points when both the first and second derivatives are zero.

Areas of Agreement / Disagreement

Participants express disagreement regarding the interpretation of the second derivative test at x = 0. Some believe it fails to provide conclusive information about the critical point, while others argue it merely suggests the possibility of an inflection point without definitive conclusions.

Contextual Notes

There are unresolved assumptions regarding the implications of the second derivative being zero and the conditions under which the second derivative test applies.

vikcool812
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Does the second derivative test fail for x3 at x=0:
f'(x)=3x2 f''(x)=6x ,

for x=0,
f'(0)=0 & f''(0)=+ve ,
so it should be a point of local maxima , but it is not!
 
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f''(0) is most certainly NOT positive!
 
vikcool812 said:
Does the second derivative test fail for x3 at x=0:
f'(x)=3x2 f''(x)=6x ,

for x=0,
f'(0)=0 & f''(0)=+ve ,
so it should be a point of local maxima , but it is not!

Since f''(0) = 0 (not +ve, whatever that means), yes, the second derivative test fails. But that doesn't mean you can't determine the type of critical point by other means.
 
^i
It didn't really fail, it just hints at the possibility of an inflection point.
 
l'Hôpital said:
^i
It didn't really fail, it just hints at the possibility of an inflection point.

No, it doesn't hint at that any more than it hints at a max or min. You could have max, min, or inflection point when the first two derivatives are zero.

And it does fail as a test distinguishing max/min.
 

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