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Minimize an cost function involving an integral

  1. Jun 18, 2012 #1
    1. The problem statement, all variables and given/known data
    The objective is to minimize

    [itex]J(x) = \int y^2 (x+y)^2 dy, [/itex]

    where x is the design variable.


    3. The attempt at a solution

    I first integrate over y to get
    [itex] J(x) = \frac{x^2 y^3}{3} + \frac{y^5}{5} + \frac{2x y^4}{4}[/itex]
    Now, I differentiate over x, and solve for the minimizer, the result is

    [itex] x_{\textrm{min}} = \frac{-3y}{4}[/itex]

    My question is the following: If I look at the cost function, I feel that the minimizer should be zero. Why am I getting [tex]\frac{-3y}{4}[/itex] as the minimzer?


    Background:
    I heard somewhere that if you want to minimize

    [itex] J(x) = \int f(y) f(x,y) dy [/itex],

    where [itex]f(x)>0[/itex] and [itex]f(x,y)>0[/itex], then you can just obtain the minimizer by minimizing f(x,y). So I constructed the example above and it doesn't work. Can anyone point to a theorem that lists this kind of result?

    Thanks.
     
    Last edited: Jun 18, 2012
  2. jcsd
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