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Math Help - integral/continuety proof question..

  1. #1
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    integral/continuety proof question..

    function f(x) continues on [a,b]
    suppose that for every sub part [\alpha ,\beta ]\subseteq [a,b]
    we have \int_{\alpha}^{\beta}f(x)dx>0.
    prove that f(x)>=0 for  x\in[a,b]

    if its wrong give a contradicting example??


    i dont have a clue from here to start or how to go.
    from the given i can conclude that if the sum of all subsections gives us a positive result then
    the total sum from a to b has to positive too.inte
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    function f(x) continues on [a,b]
    suppose that for every sub part [\alpha ,\beta ]\subseteq [a,b]
    we have \int_{\alpha}^{\beta}f(x)dx>0.
    prove that f(x)>=0 for  x\in[a,b]
    Can you prove this theorem: If a function, f, is continuous on [a,b] and \left( {\exists c \in [a,b]} \right)\left[ {f(c) < 0} \right]
    then there is a subinterval on which f is non-positive?
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  3. #3
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    i dont know how to prove it
    any hint
    ??
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  4. #4
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    i tried to prove it by contradiction:
    suppose
    there is a section which has negative value
    <br />
\int_{\alpha}^{\beta}f(x)dx<0<br />
    then
    <br />
\int_{a}^{b}f(x)dx<0 <br />
    that as far as i can think of
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