Let $f:[a,b]\to\mathbb{R}$ be continuous and suppose that
$$\int_a^bf(x)dx=\int_a^bxf(x)dx=\int_a^bx^2f(x)dx=0$$
Show that there are distinct points $x_1,x_2,x_3\in(a,b)$ such that
$$f(x_1)=f(x_2)=f(x_3)=0$$
I think this requires mean value theorem for integrals. But this can only show the existence of one such point, how to see there are at least two more?
