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!
l'Hôpital said:^i
It didn't really fail, it just hints at the possibility of an inflection point.
The second derivative test for x^3 is a method used to determine the concavity and inflection points of a function. It involves taking the second derivative of the function and analyzing its sign at critical points.
The second derivative test can be used to determine whether a critical point is a local minimum or maximum. If the second derivative is positive at a critical point, it is a local minimum. If the second derivative is negative, it is a local maximum.
If the second derivative is equal to zero at a critical point, further analysis is needed to determine whether the point is a local minimum, maximum, or inflection point. This can be done by examining the sign of the second derivative on either side of the critical point.
No, the second derivative test can only be used to find local extrema. To find absolute extrema, the first derivative test or another method must be used.
The second derivative test is based on the first derivative test. It uses the information from the first derivative to determine the concavity and inflection points of a function. The first derivative test can also be used to find local extrema, but it may not always be as efficient as the second derivative test in certain cases.