TITLE: $\int_{a}^b f(x)dx=\int_{a}^b x f(x)dx=0$, how to prove there are more than two zeros in $[a,b]$
QUESTION [5 upvotes]: $f(x)$ is continuous on $[a,b]$
One zero point follows easy from Intermediate Value Theorem and $\int_{a}^b f(x)dx=0$, but similar trick does not work for more zeros.

REPLY [2 votes]: Hint
In fact, you have that $$\int_a^b f(x)(nx-m)dx=0$$
for all $n,m\in \mathbb Z$.