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Math Help - maximisation

  1. #1
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    maximisation

    I have a question which I thought was simple enough, but have somehow got lost for quite a while on it. I need to maximise the following equation with respect to the variable x. a and b are constants.


    max \ f(x) = \frac{a}{x}\left[e^{-b}\left(1+\frac{1}{b}\right)-e^{-b x}\left(x+\frac{1}{b}\right)\right]


    Making use of the quotient rule, I got the partial derivative with respect to x to be:

    <br />
\frac{\delta f(x)}{\delta x}<br />
= -\frac{a e^{-b}}{x^2}\left(1+\frac{1}{b}\right)+\left(\frac{x b^2(a e^{-b x})+ba e^{-bx}}{{x^2 b^2}}\right)+b e^{-b x} =0<br />


    Now solving for x, I thought, would be the easy part. But I've spent quite a while now just lost in the algebra. Is there any tips or hints with regards to getting the answer out?
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  2. #2
    Super Member
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    Do you have any likely values for a and b ?
    I chose arbitrarily a=1, b=2 and plotted the graph of f(x).
    The graph has a vertical asymptote at x=0 and goes to zero (from below) as x goes to infinity.
    You could see what happens for other values (of a and b).
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  3. #3
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    Thanks. Sorry, I should have stated likely parameter values in my post. Plausible values are a=10000, and b=0.05. So for that example, the max is at x=36.
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