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Math Help - simple integral questions

  1. #1
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    simple integral questions

    These questions are stumping me for some reason, they seem simple yet I cant figure it out:
    Let f,g be riemann integrable such that int(a,b)(f) <= int(a,b)(g)
    where int(a,b) is the integral from a to b.
    is it true that f(x) <=g(x) for all x in [a,b]?
    is it true that there exists c in [a,b] such that f(c) <= g(c)?
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  2. #2
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    Take f(x)= \frac{1}{2(b-a)} and g(x)= \frac{2}{b-a} if x\in \left[ a,a+\frac{b-a}{2} \right] and g(x)=0 otherwise then \int_{a}^{b} f=1/2 \leq 1=\int_{a}^{b} g but f\nleq g.

    For the second one assume f(x) > g(x) for all x and use that the integral is monotonic (ie. preserves inequalities)
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