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Let $f:[a,b]\rightarrow R$ be a continiuous function. If $\int_{a}^{b} f(x) dx \geq 0$ is $f(x)\geq0, \forall x \in[a,b]$? I know that if $\int_{a}^{b} f(x) dx > 0$ then $f(x)\geq0$ is false, but what about the inquality that also has the equals? Is it true or false?

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    $\begingroup$ If the integral is strictly positive it is also positive. Therefore just take the same counter-example for the statement which you know is false. $\endgroup$
    – Yanko
    Mar 29 '20 at 11:19
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I think not.

$\int_{0}^{2\pi}\sin{x}dx=0\geq 0$ but certainly not $\sin{x}\geq 0$.

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