Critical points of function in 3 variables

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Homework Help Overview

The discussion revolves around identifying critical points of a scalar function in three variables, specifically analyzing the function T(r) = 2x² - 4x + y² + 4y - 3z². The original poster has identified a critical point at (1, -2, 0) and seeks to determine its nature (minimum, maximum, or saddle point).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the necessity of examining the eigenvalues of the Hessian matrix at critical points to classify them. There is mention of the Hessian matrix being composed of second partial derivatives, and some participants express familiarity with methods for functions of two variables but seek clarification for three variables.

Discussion Status

The discussion is ongoing, with participants providing insights into the classification of critical points using the Hessian matrix. The original poster acknowledges the input and indicates an understanding of the concepts discussed.

Contextual Notes

There is an indication that the original poster is more familiar with methods applicable to functions of two variables, which may influence their approach to the problem in three dimensions.

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Homework Statement


I have the scalar function
[tex]T(r)= 2x^{2} - 4x +y^2 +4y-3z^{2}[/tex] (r is obviously a vector)
I have the critical point (1,-2,0) but I'm not sure how to work out if its a min/max/saddle. I'm familiar with doing it for functions in 2 variables.
 
Last edited:
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No, for functions of 3+ variables, you have to look at the eigenvalues of the Hessian matrix at the critical points.

If all eigenvalues are positive, then crit. pt. is a minimum. If all negative, then crit. pt. is a maximum. If some of the eigenvalues are negative while the rest are positive, then the crit. pt. is a saddle point. If any of the eigenvalues are equal to zero, then our test is inconclusive.
 
Oh, and the Hessian matrix is the nxn matrix of 2nd partial deriv's of your function, where n is the number of variables (3 in your case). Look it up on wiki if you're unfamiliar.
 
Ok thanks. I edited the question while you were replying, but I understand. To anyone else this will look a mess.
 

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