# If $\int_{a}^{b} f(x) dx \geq 0$ is $f(x)\geq0, \forall x \in[a,b]$?

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?

• 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. Mar 29 '20 at 11:19

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