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Math Help - practical differential problem

  1. #1
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    practical differential problem

    having some trouble with this, but I dont know what else goes in the equation besides XY=-12, can someone help?

    "Find two numbers whose product is -12 and the sum of whose square is a minimum."

    i mostly dont know what to do on the last part
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by Arez View Post
    having some trouble with this, but I dont know what else goes in the equation besides XY=-12, can someone help?

    "Find two numbers whose product is -12 and the sum of whose square is a minimum."

    i mostly dont know what to do on the last part
    Are you tying to mininize x^2+y^2 given XY=-12?
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  3. #3
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    Quote Originally Posted by Arez View Post
    having some trouble with this, but I dont know what else goes in the equation besides XY=-12, can someone help?

    "Find two numbers whose product is -12 and the sum of whose square is a minimum."

    i mostly dont know what to do on the last part

    xy= -12 => y=-12/x
    x^2 + y^2 =z
    x^2 + (-12/x)^2=z
    differentiate z with respect to x.
    dz/dx = 2x - 288/(x^3)
    Now consider minimization criteria,
    z'=0 and z''>0 (z'=dz/dx and z'' is second derivative).
    Hence,
    For z'=0 we get,
    x^4 = 144
    Solve this for x.

    Now z''=2 + 864/(x^4)
    This is always positive, so lets take x= + sq.root(12),
    then y= - sq.root(12), or
    (x, y)= (+ sq.root(12), - sq.root(12) ).

    The other three solutions for (x,y) are,

    (- sq.root(12), + sq.root(12) )
    (+ sq.root(-12), - sq.root(-12) )
    (- sq.root(-12), + sq.root(-12) )
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  4. #4
    MHF Contributor matheagle's Avatar
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    I'm still not sure what the orginal question was.
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  5. #5
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    Quote Originally Posted by matheagle View Post
    I'm still not sure what the orginal question was.
    I think it meant find two numbers whose product is -12 and the sum of those two numbers is minimum.
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  6. #6
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    Quote Originally Posted by matheagle View Post
    I'm still not sure what the orginal question was.
    I suspect it's (x + y)^2 that is to be minimised ....
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  7. #7
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    Quote Originally Posted by niranjan View Post
    I think it meant find two numbers whose product is -12 and the sum of those two numbers is minimum.
    I meant sum of the squares of those numbers to be minimum i.e.,
     x^2 +y ^2
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  8. #8
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by akshatha View Post
    I meant sum of the squares of those numbers to be minimum i.e.,
     x^2 +y ^2

    I figured that
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  9. #9
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    Quote Originally Posted by akshatha View Post
    I meant sum of the squares of those numbers to be minimum i.e.,
     x^2 +y ^2
    Yes, I mean the same. I quoted wrongly that its sum of numbers, but its sum of squares of numbers which has to be minimized. I did math work for squares, but misquoted it.
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