Level Sets and Degenerate Critical Points

In summary, the conversation discusses the concept of degenerate critical points of a function and how they are related to the level set of another function. It is noted that the question is vague and lacks necessary information for a comprehensive answer.
  • #1
autarch
9
0
How would one show that if there is a number c for which g'(c)=0, then every point on the level set {(x,y)|H(x,y)=c} is a degenerate critical point of f?


I know that the question may seem vague, but this is the question as it was given to me by my professor. It is something to think about, but. I would appreciate a comprehensive answer; however, any comments would also be appreciated.
 
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  • #2
it is not only vague but so vague as to be entirely meaningless. None of the relevant functions are defined or related to each other.
 
  • #3
mathwonk said:
it is not only vague but so vague as to be entirely meaningless. None of the relevant functions are defined or related to each other.

Now you know how I feel.
 

1) What are level sets in mathematics?

Level sets are a collection of points in a function's domain that map to the same output value. In other words, they are the set of all points where the function has a constant value.

2) How are level sets useful in understanding functions?

Level sets provide a visual representation of how a function behaves. By plotting level sets, we can see the shape and behavior of the function, as well as identify any critical points or regions of interest.

3) What is a degenerate critical point?

A degenerate critical point is a point on a level set where the function does not have a well-defined tangent plane. This can occur when the function's gradient is zero or undefined at that point.

4) How do degenerate critical points impact the behavior of a function?

Degenerate critical points can cause complications in determining the behavior of a function. They can create discontinuities or change the direction of the function's gradient, making it difficult to analyze or optimize the function.

5) How are level sets and degenerate critical points used in real-world applications?

Level sets and degenerate critical points are commonly used in fields such as physics, engineering, and computer science. They are used to model and analyze complex systems, optimize functions, and solve real-world problems in various industries.

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