Let $f,g$ be differentiable functions such that $\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$. Show that $g(0)=0$

Question

Let $f,g$ be differentiable functions such that $$\int\limits_0^{f(x)}f(t)g(t)dt=g(f(x))$$ Show that $g(0)=0$

I know it is done if $f(x)=0$ for some $x$, but I just have $$f(f(x))g(f(x))f'(x)=g'(f(x))f'(x)$$ by FTC, but then I dont know what to do.

Answer 1

Sorry, i don't understand. What about $f(t)=1$ and $g(t)=1$ for every $t$? Maybe there are other hypothesis.