Distinguishing Max, Min & Saddle - Choosing Coordinate Systems

  • Thread starter MarianaA
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In summary, the conversation discusses two questions: how to distinguish between a maximum, minimum, or saddle point, and when to use a particular coordinate system (rectangular, cylindrical, or spherical). The suggested approach for distinguishing between the different types of points is by looking at the Hessian and its eigenvalues. As for the second question, it is recommended to use cylindrical coordinates for systems with axis symmetry and spherical coordinates for systems with point symmetry. For all other cases, rectangular coordinates are suitable. It is also mentioned that this is not a homework question, but rather a genuine inquiry.
  • #1
MarianaA
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Hi every one, I have two questions, I hope some one can help me.
Thanks.

How can I distinguish between a maximum, minimum or a saddle?

When are times when a particular coordinate system is a better choice the n the other two (rectangular, cylindrical and spherical)?
 
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  • #2
How can I distinguish between a maximum, minimum or a saddle?
You could start by writing out the definition for each.
 
  • #3
but I don't know the definition
 
  • #4
A minimum occurs when the Hessian is positive definite, a maximum when it is negative definite, and a saddle point when the Hessian has positive and negative eigenvalues.
 
  • #5
Thanks ^_^
How about my second question:

When are times when a particular coordinate system is a better choice the n the other two (rectangular, cylindrical and spherical)?
 
  • #7
I am assuming this is not homework. If it is, this thread will probably be deleted.
 
  • #8
Well I need to know this so I can do my homework
Thanks ^_^
 

FAQ: Distinguishing Max, Min & Saddle - Choosing Coordinate Systems

What is the difference between a maximum and a minimum point?

A maximum point is the highest point on a graph or function, while a minimum point is the lowest point. In other words, a maximum point is where the function reaches its peak, and a minimum point is where the function reaches its lowest value.

How can I determine if a point is a maximum or a minimum?

To determine if a point is a maximum or a minimum, you must look at the slope of the function at that point. If the slope is positive, the point is a minimum, and if the slope is negative, the point is a maximum. Additionally, you can take the second derivative of the function at the point. If the second derivative is positive, the point is a minimum, and if it is negative, the point is a maximum.

What is a saddle point?

A saddle point is a point on a graph or function where the slope is zero, but it is neither a maximum nor a minimum. It is a point of inflection, where the concavity of the function changes. This means that the function is increasing in one direction and decreasing in another at this point.

How do I choose a coordinate system for distinguishing max, min, and saddle points?

The best way to choose a coordinate system is by looking at the symmetry of the function. If the function is symmetric about the x-axis, then a Cartesian coordinate system is suitable. If the function is symmetric about the y-axis, a polar coordinate system may be more appropriate. Additionally, you can choose a coordinate system based on the features of the function, such as using a logarithmic coordinate system for exponential functions.

What is the significance of distinguishing between max, min, and saddle points?

Distinguishing between max, min, and saddle points is crucial in understanding the behavior of a function. It allows us to determine the critical points of a function, which can help us find the maximum or minimum values and the intervals where the function is increasing or decreasing. This information is essential in various fields, such as economics, physics, and engineering.

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