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Math Help - Integrals problem 2

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    MHF Contributor Also sprach Zarathustra's Avatar
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    Integrals problem 2

    Let function f be continuous and non-negative in [a,b], and which holds:
    int{[a,b] f dx} = 0.

    Prove that: f(x)=0 for all x in [a,b].
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Let function f be continuous and non-negative in [a,b], and which holds:
    int{[a,b] f dx} = 0.

    Prove that: f(x)=0 for all x in [a,b].
    Well this is intuitively clear, and a hand-wavy approach would be to say that if f(x) were to rise above 0 at any point, it would make the integral greater than zero in that local region, and since f(x) is non-negative there's no way to balance out the overall integral to zero, therefore f(x) = 0 in [a,b].

    A more rigorous approach might involve the intermediate value theorem.. not sure, I'd have to think about it more.
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