Lagrange Multipliers: Advantages & Necessity?

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SUMMARY

Lagrange multipliers are a method used for finding the extrema of functions subject to constraints. In certain minimization or maximization problems, such as determining the shortest distance from a point to a plane, one can solve the problem using either Lagrange multipliers or by substituting variables and setting partial derivatives to zero. While both methods can yield the same result, Lagrange multipliers are essential for constrained optimization scenarios. Understanding when to apply Lagrange multipliers versus alternative methods is crucial for effective problem-solving in calculus.

PREREQUISITES
  • Understanding of calculus, specifically partial derivatives
  • Familiarity with optimization techniques
  • Knowledge of constrained optimization problems
  • Basic proficiency in mathematical notation and terminology
NEXT STEPS
  • Study the method of Lagrange multipliers in detail
  • Explore examples of constrained optimization problems
  • Learn about alternative optimization techniques, such as substitution methods
  • Review applications of Lagrange multipliers in real-world scenarios
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Students of calculus, mathematicians, and professionals in fields requiring optimization techniques, such as engineering and economics.

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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