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# Problems Involving Optimization of Polynomial Integrals

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Compute

min (a,b,c) the definite integral from -1 to 1 of |x^8 - a - bx - cx^2|^2 dx

and find

max the definite integral from -1 to 1 of (x^3)(g)(x) dx

where g is subject to the restrictions

the definite integral from -1 to 1 of g(x) dx =
the definite integral from -1 to 1 of xg(x) dx =
the definite integral from -1 to 1 of (x62)(g)(x) dx = 0;

the definite integral from -1 to 1 of |g(x)|^2 dx = 1

##### Solution Summary

In this solution we find the maxima and minima of various polynomial integrals using properties of Legendre polynomials in an attached Word document.

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