Stone–Weierstrass theorem exercise

Question

$f$ is continuous and $$ \int_{a}^{b} x^nf(x) dx=0 $$ for every $n\leqslant N$.

Prove that $f$ has at least $N+1$ zero points at $(a,b)$.

Answer 1

Hint: If $f$ has $N$ or fewer zero-points on $(a,b)$, then there exists a non-zero polynoimal $p(x)$ of degree $N$ such that $f(x)p(x) \geq 0$ for all $x \in [a,b]$.