For $a, suppose that $f$ is continuous on $[a,b]$ and $\int_{a}^{b}f(x)dx=0$. 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
, the equation tells us that the interval before $c $ is of opposite polarity and equal to the interval after $c$. That means the function goes from (1). either negative to positive or (2).positive to negative or the (3).function is zero. (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!