# Thread: minimize the integration value

1. ## 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!

2. 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?