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Math Help - Differentiation problem (shouldn't be too hard)

  1. #1
    roblomas
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    Differentiation problem (shouldn't be too hard)

    Hi, I have two equations that i need to find the max of...apparently this is finding the derivative of the function..but I'm struggling..

    1.

    find max of: a(1-X)^1/2 + (X)^1/2

    where a is a positive constant.
    So I need to find out what X equals.

    2.

    find max of: 4log(100-X) + 2log(X + Y)

    So I need to find X.

    Any help would be much appreciated!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by roblomas View Post
    Hi, I have two equations that i need to find the max of...apparently this is finding the derivative of the function..but I'm struggling..

    1.

    find max of: a(1-X)^1/2 + (X)^1/2

    where a is a positive constant.
    So I need to find out what X equals.
    At a local maximum of a differentiable function f we have:

    df/dx=0.

    In this case:

    f(x) = a(1-x)^1/2 + (x)^1/2

    so:

    df/dx= a(1/2)(1-x)^{-1/2}(-1) + (1/2) x^{-1/2}

    so if df/dx=0 we have:

    x^{-1/2} = a (1-x)^{-1/2}

    squaring:

    1/x = a^2/(1-x)

    and if x!=0 and x!=1 (you can check neither of these give df/fx=0 so this
    is OK), we have:

    x=(1-x)/a

    so x=1/(1+a)


    Now this can be a maximum, a minimum or a point of inflection so we need to
    check that this is a maximum using the second derivative test.

    d^2f/dx^2 = -a/[4(1-x)^(3/2)] - 1/[4x^{3/2}]

    which is negative at x=1/(1+a) (as a>0), hence this is a maximum.

    RonL
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