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D-admissible directions and extermal points

  1. Feb 5, 2008 #1
    Let X = C[a,b], J(y) = integ(a to b) sin^3(x) + y(x)^2 dx and D={yEX; integ(a to b) y(x)dx = 1}
    (a) what are the D-admissible directions for J?
    (b) Find all possible (local) external points for J on D?

    so far i have:
    (let e be epsilon)
    lim e->0 J(y+ev) - J(y) / e
    = lim e->0 integ (a to b) [ sin^3(x) + (y+ev)^2(x)dx - J(y) ] / e
    = lim e->0 integ (a to b) [2ey(x)v(x) + e^2v(x)^2) dx ] / e
    = 2 integ(a to b) y(x)v(x) dx
    A similar example was done in class, so i just copied it for this question.
    G(y) = integ(a to b) y(x) dx = 1
    gateaux G(y;v) = integ (a to b) y(x)v(x) dx

    is this right so far? how do i go about answering (a) and (b).
  2. jcsd
  3. Feb 6, 2008 #2
    i was wondering if the D-admissible directions for J is just
    y+Ev E D
    integ (a-b) (y+Ev)(x) dx
    = 1+ E integ(a-b) v(x) dx

    can i simplify any further?
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