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Math Help - find x and z which maximizes..

  1. #1
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    find x and z which maximizes..

    does anyone know how to do this

    find x and z that maximizes ln(4-x) + ln(4+z) + ln(z)

    subject to

    2 lnx + ln(4-z) + ln(4-z) = 2ln(32/7)

    anyhelp is much appreciated
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chogo View Post
    does anyone know how to do this

    find x and z that maximizes ln(4-x) + ln(4+z) + ln(z)

    subject to

    2 lnx + ln(4-z) + ln(4-z) = 2ln(32/7)

    anyhelp is much appreciated
    the values of x and z that maximise ln(4-x) + ln(4+z) + ln(z), also
    maximise:

    (16-x^2)z ...(1)

    as x>0, the constraint may be rewritten:

    x^2(4-z)^=(32/7)^2,

    or:

    x=32/(7(4-z)) ...(2)

    So substitute (2) into (1) and find the z that maximises this now
    unconstrained objective, find the corresponding x from 2, and substitute
    back into ln(4-x) + ln(4+z) + ln(z) to find its maximum.

    RonL
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  3. #3
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    thank you very much
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  4. #4
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    im sorry to be a pain but i seem to be very thick and dont understand how you obtained equation (1)

    (16-x^2)z ...(1)

    if you have the time could you please explain this to me
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by chogo View Post
    im sorry to be a pain but i seem to be very thick and dont understand how you obtained equation (1)

    (16-x^2)z ...(1)

    if you have the time could you please explain this to me
    You are looking for the maximum of g(x,z)=ln(4-x) + ln(4+z) + ln(z), but:

    g(x,z)=ln( (4-x)(4+x)x )

    by the law of logarithms.

    Then let:

    f(x,z)=exp(g(x,z))=(4-x)(4+x)x,

    and because the exponential function is increasing on the real numbers, any
    maxima of g(x,z) is also a maxima of f(x,z) and vice-versa.

    RonL
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