Determining integration and continuity

Reference no: EM13129123

1. Prove that if g(x) is nonnegative and continuous on [0, 1] and (integral from 0 to 1 of g(x)dx) = zero then g(x)= 0 on [0,1]

2. If f is continuous on [0,1] and if (integral from zero to one of (f(x) x^n)dx)) =0 for n in the Naturals, prove that f(x)=0 on [0,1]/ Hint: The integral of the product of f with any polynomial is zero. Use the Weierstrass approximation theorem to show that (integral from 0 to one of f^2 (x) dx) =0. I think f^2(x) =(f(x))^2 here.

Reference no: EM13129123

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