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Math Help - minimize the integration value

  1. #1
    Senior Member Shanks's Avatar
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    minimize the integration value

    (1)Find real numbers a,b,c\text{ such that they minimize the following integration }:
    \int_0^\infty \left|x^3-a-bx-cx^2\right|^2e^{-x}dx

    (2)Find real numbers a,b,c\text{ such that they minimize the following integration }:
    \int_0^\infty \left|x^3-a-bx-cx^2\right|e^{-x}dx .

    The first problem is easier than the second one. If somebody posts the elaborate correct solution for the second one, I will be very appreciate!
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  2. #2
    Senior Member Shanks's Avatar
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    for the first problem, we can consider it as Projection:
    find the distance of a point and a given subspace in Hilbert space.
    we can do it by using the Projection Theorem in Hilbert space.
    for the second problem, Since L^1\left[0,\infty\right] is not a hilbert space, we can't apply the theorem.
    Is There any other way to solve it?
    auguement on a,b,c?
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