Let $f$ be a continuous function on $[0,1]$ satisfying $\int_0^1f(x)\,dx = 0$
and $\int_0^1xf(x)\,dx = 0.$ Show that there exists $a$,$b$ in $[0,1]$ with $a < b$, such that $f(a) = 0 =f(b)$. Existence of one point is clear to me but I cannot prove the existence of the other one.
Thanks for any help.