For $a$\int_{a}^{b}f(x)dx=0$. Prove that there is at least one number in $(a,b)$ such that $f(c)=0$.
I attempted this
Let $c$ be any number in $(a,b)$
$\int_{a}^{b}f(x)dx=0$ implies $\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx=0$ implies $\int_{a}^{c}f(x)dx=-\int_{c}^{b}f(x)dx$
Since $a
(3) If the function is zero, we are done as f(c)=0.
(1),(2) That is, either $f(a)>0 $ and $f(b)<0$ or $f(a)<0 $ and $f(b)>0$, and since $f(x)$ is continuous, by the Intermediate Value Theorem, there exists a $c$ such that
$$f(c)=0$$
I am very unsure if I would get my marks for showing this. Is it a correct proof? I am also looking for more elegant proofs. Thanks in advance!