TITLE: If $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$
QUESTION [5 upvotes]: Assume $f$ is continuous on $[a,b]$, if $\int_a^b f(x)\,dx=0$, prove that $f(c)=0$ for at least one $c$ in $[a,b]$.

The problem didn't state anything about the function $f$, is it safe to assume either:

$f$ is an odd function and implies that there is some $x_1$ $x_2$ in $[a,b]$ such that $f(x_1)<0$, $f(x_2)>0$ and apply Bolzano's Theorem to conclude that there is at least a $c$ in $[a,b]$ such that $f(c)=0$.

$f$ is $0$ for all $x$ in $[a,b]$ hence it is trivial.


Is this argument correct?

REPLY [8 votes]: By first mean value theorem for integration we have that exists $c\in\left[a,b\right]$
  such that (assuming $a<b$
 ) $$0=\int_{a}^{b}f\left(x\right)dx=f\left(c\right)\left(b-a\right)$$
 then $$f\left(c\right)=0.$$