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Math Help - Help with Lagrange Multipliers

  1. #1
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    Question Help with Lagrange Multipliers

    I am new to the forums, so please forgive any errors in formatting a request for help. Thanks to everyone who views this!
    --------------------------------------------------------------------------------------------------------------------
    Here is the question:
    "Use Lagrange multipliers to find the point (a, b) on the graph y=ex , where the value ab is as small as possible."

    I am unsure how to solve this. Any help would be greatly appreciated since I would like to actually understand the solution and how it is solved!

    Once again, thank you!
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  2. #2
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    Re: Help with Lagrange Multipliers

    Hello, Confucius!

    Use Lagrange multipliers to find the point (x, y) on the graph y=ex , where the value xy is as small as possible.

    We want to minimize: f(x,y) \,=\,xy subject to the constraint: e^x - y \,=\,0

    We have: . F(x,y,\lambda) \;=\;xy + \lambda(e^x-y)

    Set the partial derivatives equal to zero, and solve.

    . . \begin{array}{cccccccc}F_x &=& y + \lambda e^x &=& 0 & [1] \\ F_y &=& x - \lambda &=& 0 & [2] \\ F_{\lambda} &=& e^x-y &=& 0 &[3] \end{array}


    From [2]: . \lambda \,=\,x

    Substitute into [1]: . y + xe^x \:=\:0 \quad\Rightarrow\quad y \:=\:-xe^x

    Substitute into [3]: . e^x - (-xe^x) \:=\:0 \quad\Rightarrow\quad e^x + xe^x \:=\:0

    . . . . . . . . . . . . . . . . . e^x(1+x) \:=\:0 \quad\Rightarrow\quad x \:=\:-1

    Substitute into [3]: . e^{-1} - y \:=\:0 \quad\Rightarrow\quad y \:=\:\tfrac{1}{e}


    Therefore: . (x,y) \;=\;\left(-1,\,\frac{1}{e}\right)
    Thanks from Confucius
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  3. #3
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    Re: Help with Lagrange Multipliers

    Thank you very much!
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