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Math Help - Lagrange method of constrained optimization question

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    Lagrange method of constrained optimization question

    Lagrange method of constrained optimization question-screen-shot-2013-05-15-1.19.30-am.png

    in the picture above, shouldn't the last expression be the negative of what it is? Shouldn't it be

    (-1)( p1x1 + p2x2 - m ) = -p1x1 - p2x2 + m ?
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    MHF Contributor MarkFL's Avatar
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    Re: Lagrange method of constrained optimization question

    Yes, but since it is equated to zero, they have simplified by multiplying through by negative 1, I am guessing so that there is only 1 negative term, rather than two.
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    Re: Lagrange method of constrained optimization question

    Does that mean I could do the same (multiply by -1) to the first two partial derivatives and not change the answer set either?
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    MHF Contributor MarkFL's Avatar
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    Re: Lagrange method of constrained optimization question

    Yes, algebraically there would be no problem in doing that if you wish. I was taught the equivalent notation:

    If given f\left(x_1,x_2 \right) subject to the constraint g\left(x_1,x_2 \right)=p_1x_1+p_2x_2-m=0, then we wish to solve the following system:

    f_{x_1}\left(x_1,x_2 \right)=\lambda\cdot g_{x_1}\left(x_1,x_2 \right)

    f_{x_2}\left(x_1,x_2 \right)=\lambda\cdot g_{x_2}\left(x_1,x_2 \right)

    p_1x_1+p_2x_2-m=0

    Then we draw an implied relationship between the two independent variables from the first to equations which we then use in the third to get the critical point(s).
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