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Math Help - Multi-Variables Calculus [Max,Min,Lag.Multipliers]

  1. #1
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    Multi-Variables Calculus [Max,Min,Lag.Multipliers]

    Hey there,

    I realy need some help in the following question:
    Find absolute maximum and minimum of the function:
    f(x,y,z)=e^{-2x-3y-5z} in the set:
     {(x,y,z) | x^{2}+y^{2}+z<=1,x+y+z>=1 } .
    {Does the fact the the function \phi (t) =e^{t} is monotonic can help? } ...


    Well, we know that the only place where the partial deriatives are zero is (0,0,0)... But it doesn't tell us anything about absolute min/max... I'll be delighted to get some help on how to continue...


    Thanks!
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Hey there,

    I realy need some help in the following question:
    Find absolute maximum and minimum of the function:
    f(x,y,z)=e^{-2x-3y-5z} in the set:
     {(x,y,z) | x^{2}+y^{2}+z<=1,x+y+z>=1 } .
    {Does the fact the the function \phi (t) =e^{t} is monotonic can help? } ...


    Well, we know that the only place where the partial deriatives are zero is (0,0,0)... But it doesn't tell us anything about absolute min/max... I'll be delighted to get some help on how to continue...


    Thanks!
    You'll notice that

    \left[\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right] = \left[-2e^{-2x - 3y - 5z}, -3e^{-2x - 3y - 5z}, -5e^{-2x - 3y - 5z}\right].

    Since the exponential function is always >0, that means there are not any stationary points in this region.

    So the absolute minimum and maximum must be somewhere on the boundaries.
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  3. #3
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    Yep...You're right...I've managed to get to this point too (my statement about (0,0,0) is wrong indeed...) ... But my problem is I can't figure out how to find the absolute min/max and how to prove they're absolute min/max ....
    The boundaries of this region are
    {(x,y,z) | x^{2}+y^{2}+z=1,x+y+z=1 } ...How can I find the absolute min/max in this region?


    Hope you'll be able to help me


    Thanks a lot!
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