Determine the continuous functions $ f:[0,\infty ]\rightarrow \mathbb{R}$, for which $\int_{x}^{y}xf(t)dt=\int_{1}^{y}f(t)dt,\forall x,y\in (0,\infty ) $.
Here's what I did.
Let $F$ be a primitive of $f$.
$\int_{x}^{y}xf(t)dt=\int_{1}^{y}f(t)dt\Leftrightarrow xF(y)-xF(x)=F(y)-F(1).$
If we put $y=1 $ we obtain that $xF(1)-xF(x)=0$, so $ F(x)=F(1)=c,c\in \mathbb{R}$, which means that $F$ is a constant function. Hence, $f(x)=0,\forall x\in (0,\infty )$.
Is my approach correct?