For which functions is $\sqrt{\int_1^2 f(x)^2*xdx} = 0$ true

Question

I have to find all functions for which the following statement is true $$\sqrt{\int_1^2 f(x)^2*xdx} = 0$$ $f(x)$ can be anything: polynomial, trigonometric, exponential, ... I know I can leave the square root and I have to find an antiderivative for which $F(2)-F(1) = 0$ is true, but I can't seem to get any further than that... any tips or answers would help a lot!

Answer 1

$f(x)$ has to be identically zero namely $f(x)=0$., because the integrand of the area-integral $A=\int_{1}^{2} x f^2(x) dx$ is positive definite in $(1,2)$. Hence $A$ cannot be zero and if it is zero then $f(x)=0$.

Answer 2

Observe that $x>0, f(x)^2\ge0$. Therefore for the integral to be $0$ you need $f(x)=0$ almost everywhere on $[1,2]$.