Need hint for integral question

Question

Is it possible to show that $$ \int_a^b f(x)^2g(x) dx = 0 $$ if we know that $$ \int_a^b f(x)dx = \int_a^bg(x)dx = 0 $$

This is a step in a bit of a longer problem, but I got stuck at this point. In my eyes it would appear that since we are squaring $f(x)$ we will get a fully positive domain but I don't think that any arbitrary function that satisfies this property can be enough to make their product integral $0$. Any hints to show its truth or counterexample would be appreciated!

Answer 1

Let $g$ be a broken line joing $(-4,0),(-2,2),(0,0),(1,-2),(4,0)$ and $f(x)=x$. Integrate on $[-4,4]$.

Answer 2

Here's a picture to give you some intuition that this is wrong: