Lagrange Multipliers: Advantages & Necessity?

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Homework Help Overview

The discussion revolves around the method of Lagrange multipliers, particularly in the context of optimization problems involving minimization and maximization, such as finding the shortest distance from a point to a plane. Participants explore the necessity and advantages of using Lagrange multipliers compared to alternative methods.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the possibility of solving optimization problems both with and without using Lagrange multipliers. Questions are raised about the necessity of the method and how to determine when it is more advantageous to use it.

Discussion Status

The discussion includes various perspectives on the use of Lagrange multipliers, with some participants suggesting that it is not always necessary, while others emphasize its importance for constrained optimization. There is an acknowledgment of different approaches to the same problem, but no explicit consensus has been reached.

Contextual Notes

Participants are considering the implications of substituting variables in optimization problems and the timing of applying Lagrange multipliers. There is an indication that examples may be beneficial for further clarification, but specific examples have not been provided in the discussion.

DR13
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I do not have one specific question that needs answering. Rather, it is about Lagrange multipliers in general.

So for certain minimization/maximization questions (ie find the shortest distance from some point to some plane) it seems that one could solve the question using lagrange multipliers and not using lagrange multipliers (I have done it both ways). One could set both partials equal to 0 and solve for x and y without using lagrange (as long as a function of x and y is substitued for z). Or, one could not substitute for z and use lagrange multipliers to find the distange.

Is this right? Or am I missing something? Also, is it ever 100% necessary to use lagrange multipliers? How can one tell when it is better to use lagrange multipliers and when it is better not to?

Thanks
DR13
 
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Ok, you do the same steps required by Lagrange multipliers, but you don't call it Lagrange multipliers. :)

Using Lagrange is the only way for constrained maximum/minimum.
 
Oh I get it. By substituting for z before taking the partials you are just doing the step earlier rather than later. Is that right?
 
I'm not sure I follow you completely.
Do you have an example ?
 
Its fine. I worked it out myself and get it now. Thanks!
 

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