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Find the extremum of z = xy subject to x + y = 6 Watch

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    Please help with the following: Find the extremum of z = xy subject to x + y = 6

    By substituting from the constraint into the objective function.
    And by Lagrange's method..

    Thanks
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    (Original post by JKITFC)
    Please help with the following: Find the extremum of z = xy subject to x + y = 6

    By substituting from the constraint into the objective function.
    And by Lagrange's method..

    Thanks
    I suggest you learn about Lagrange multipliers before, since this is a basic application that comes from learning
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    Use the AM-GM inequality to find an upper bound for (xy)^0.5, hence xy
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    (Original post by 123Master321)
    I suggest you learn about Lagrange multipliers before, since this is a basic application that comes from learning
    I know it... I cant do the systems of equations that it produces
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    (Original post by JKITFC)
    I know it... I cant do the systems of equations that it produces
    Ok, so:
    

 f(x,y)=xy

g(x,y)=6 \implies g_2(x,y)=x+y-6=0
     \mathcal{L} =f(x,y) +\lambda g_2(x,y)
    \mathcal{L} =xy +\lambda (x+y-6)
    \frac{ \partial \mathcal{L}}{\partial x}= y+ \lambda =0
    \frac{\partial \mathcal{L}}{\partial y} = x+ \lambda =0
    y=-\lambda

x=-\lambda

Since x+y=6
     \lambda =-3 \implies x=y=3
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    (Original post by MarcusRashford19)
    Use the AM-GM inequality to find an upper bound for (xy)^0.5, hence xy
    OP was asking for using Lagrange multipliers, otherwise you could just let y=6-x, put it into the other equation and differentiate, no need for AM-GM
 
 
 
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