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Math Help - Lagrange Multipliers and Inequalities

  1. #1
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    Lagrange Multipliers and Inequalities

    Is there a way to apply the method of Lagrangian Multipliers when your equation of constraint is an inequality(it won't be called an equation of constraint then I guess). Here is an example of what I came up with.

    Say f(x)=x^2

    You want to find the extrema satisfying the inequality -1\le x\le1

    If you introduce a variable z that satisfies the equation x^2+z^2=1 that means x\in[-1,1]

    g(x,z)=x^2+z^2=1 becomes your equation of constraint. Now we have

    \nable f=\lambda\nabla g where \nabla =\left(\dfrac{\partial}{\partial x},\dfrac{\partial}{\partial z}\right)

    We get \left(2x,0\right)=\lambda\left(2x,2z\right)

    From 2\lambda z=0 we get z=0 or \lambda=0

    From 2x=2\lambda x we get x=0 or \lambda=1

    For x=0 we get our minimum.

    For \lambda=1 we have z=0 and x^2=1\therefore x=\pm 1 which is the local maximum.

    Now what I want to know is, if there is a way to solve for more general inequalities like x<3 or x>7.

    Any input will be appreciated.
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  2. #2
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    The First Dimension

    In principle, I think the answer is yes, by introducing the simple disk as you have, which was a good idea. So for x<3, you would let g(x,z)=x^2+z^2=9 . For x>7, the domain is a bit trickier, but you can replace with \frac1x<\frac17 and make g(x,z)=\frac1{x^2}+z^2=\frac1{49} to get x>7 or x<-7. I think this might be what you are looking for.

    However, keep in mind that in one dimension, there is little point to Lagrange multipliers. Consider: if \nabla f = \lambda \nabla g, then by letting \lambda=0, \nabla f=0, so (\frac{\partial}{\partial x}, 0)=(0,0), i.e. f'(x)=0, which is the standard one-dimensional way of finding the maxima/minima anyway, having nothing to do with g(x,z).

    You might try experimenting with your method on functions f(x,y) over two variables to see if there is something truly new to be discovered.
    Last edited by Media_Man; July 24th 2010 at 07:39 AM. Reason: the voices told me to
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