Is $f$ that satisfies $\int_a^bf(x)dx=\int_a^bxf(x)dx=\int_a^bx^2f(x)dx=0$ a constant? My question is:
Define $f\in C^{0}[a,b]$ satisfying $\int_a^bf(x)dx=\int_a^bxf(x)dx=\int_a^bx^2f(x)dx=0$. Must this function be a constant function?
I made a easy analysis, which can prove that $f$ has at least three points, as follows

we have $\int_{a}^{b}g(x)f\left(x\right)dx=0$ , in which $g(x)=mx^2+nx+p$. $\forall m,n,p \in \mathbb{R}$
If $f$ has less than three zero points, we can select appropriate $m$,$n$,$p$ satisfying  $g(x)f(x)>0$ for all $x$ which is not the zero point of $f$. And it's contradictory to $\int_{a}^{b}g(x)f\left(x\right)dx=0$. Thus $f$ must have at least three zero points.

So can we make a further analysis that indicates $f$ is a constant function?Any ideas would be highly appreciated!

The constraints on $f$ only mean that $f$ is orthogonal to the vector space of polynomials of degree less than 2. But there do exist loads of functions who are not constants and satisfy this conditions.
Take a look at the Gram Schmidt orthonormalisation procedure and try to find a polynomial (for example of degree 3) orthogonal to the polynomials $1$, $x$ and $x^2$.
