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Problem Involving Maximizing the Ratio of Integrals

  1. Mar 7, 2012 #1
    The ratio of integrals

    ∫〖a(x) b(x)dx〗/ ∫c(x) b(x)dx

    can be maximized by choosing b(x) equal to the delta function at the point where a(x)/c(x) is a maximum.

    Can anyone provide the solution for choosing b(x) when b(x) cannot equal the delta function, b(x) is greater than zero with a maximum value of C, and a(x) and c(x) are both positive over the integration interval and also monotonically increasing?
  2. jcsd
  3. Mar 7, 2012 #2


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    Science Advisor

    I suspect the best you could do is take an interval around the x where a(x)/c(x) is maximum. The width of the interval would be 1/C, while b(x) = C in the interval, and = 0 otherwise.
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