If a function is continuous on an open interval, and at a given point its first derivative is zero and the second derivative is zero, is that point an absolute max/min? Should the first derivative be zero and second derivative be non-zero at a point where the absolute max/min exists, and vice versa?
2026-04-01 13:29:02.1775050142
On
Global extremum on an open interval
1.2k Views Asked by user711493 https://math.techqa.club/user/user711493/detail At
2
There are 2 best solutions below
1
On
Not necessarily it is maximum or minimum for details please check the link https://www.mathsdiscussion.com/maximum-and-minimum-of-a-function/
From the comments, it's obvious that equating 1st derivative to zero is inconclusive for finding the extrema. It is a necessary condition for there to be an extrema, but it is not always there, that is, an extremum will always have 1st derivative zero but not all points with zero first derivative are max/min. Inflection points, undulation points etc come into the picture.
The main thing to observe with second derivatives considering extrema is that the sign remains same in the vicinity of an extremum (this is so because the curvature remains same). It may or may not be zero but if the sign remains unchanged, then it is an extremum. For if it happens otherwise, then it may be an inflection point etc. You can try this with examples.