Let f and g be continuous functions defined on a compact interval [a, b]. Assume that g(x) > 0 for all x E [a, b]. Prove that there exists xo E [a, b] such that qu | f(x)g(x)dx = f(wo) / g(x)dx. a (Hint: consider using change of variable and the integral mean value theorem or the second mean value theorem.) Is the same conclusion true if we remove the assumption g(x) > 0? If yes, please give a proof; if no, please give a counterexample.

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