What are the key concepts of Lagrange Multipliers?

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Lagrange Multipliers are used to find the local maxima and minima of a function subject to constraints. The technique involves introducing an extra variable, the Lagrange multiplier, which represents the rate of change of the objective function with respect to the constraint. Understanding the constraints is crucial, as they define the feasible region for the optimization problem. The discussion emphasizes the need for a graphical interpretation of the function rather than its applications in fields like economics. Clarification on partial differentiation and gradients in multiple dimensions is also sought for a better grasp of the concepts.
mtanti
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I've just started multi dimensional calculus, among which Langrange's Multipliers. I have some questions which will help me grasp the concepts since I'm a very curious guy...

a) What are you finding exactly with this technique?
b) What is the constraint?
c) What does the extra variable represent?
d) Is there another more intuitive but less effiecient way to obtain the same result?

My lecturer is a very poor teacher and just wants to get on with the syllabus I guess... I would like an explanation on the actual graph of the function and not about the practical usage of it such as in economics...

P.S. I understood the reasoning behind partial differentiation but I'm still shakey on the fact that to differentiate a plane you can only find a general gradient for each axis (y and x) and not a single general gradient function f'(x,y).

Thanks!
 
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