Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2023
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18
Types of critical points Let us now consider that there are more than maximum and minimum. Therefore,remember that we considered the example of x2 + y2, which has a critical point butis obviously a minimum. Of course this could be a local minimum because it could bethat in a more complicated function there is indeed a minimum here. But elsewhere, the function drops to a lower value. So we call that just a localminimum to say that it's a minimum if you stick to values that are close enough tothat point.
The function also has a local maximum, but it is easy to locate because it occurswhere the curve becomes steeper. However, there is another example of a critical point: the saddle point. It is aphenomenon that cannot be observed in single variable calculus, but it does exist.Here, we see the critical point at the origin is a saddle point because depending onwhich direction we look from it will either appear to be a maximum or minimum. If you look at the tangent plane to this graph, you'll see that it is horizontal at theorigin. You have a mountain pass, where the ground is horizontal. But depending onwhich direction you go, you go up or down. So we say that if a point is neither aminimum or maximum, it's a saddle point.
Types of Critical Points
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