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

[vikcool812]

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!

 

[jbunniii]

f''(0) is most certainly NOT positive!

 

[LCKurtz]

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.

 

[l'Hôpital]

^i
It didn't really fail, it just hints at the possibility of an inflection point.

 

[LCKurtz]

^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.