Definition of Critical Points
Recall for 1 variable functions, a critical point occurs at a point if or does not exist. The concept is the same for 2 variable functions, except we must modify a few things.
The point is a critical point of if one of the following is true:
Types of Critical Points
- and/or does not exist
There are 3 types of critical points:
Classifying Critical Points
- Local Maximum: occurs when
for all points that is around . In other words, it's the biggest value of the function around it's region.
- Local Minimum: occurs when for all points
that is around . In other words, it's the smallest value of the function around it's region.
- Saddle point: neither a local minimum or local maximum.
is a critical point of
. To see whether it's a local maximum, or local minimum, or saddle point, we compute the following:
- and , then it is a local minimum
- and , then it is a local maximum
- , then it is a saddle point
- , then it could be any of the 3 types. Need to use other techniques to classify it.