We are given that $\vec{F}=k\left<y,x,0\right>$, and asked whether $\vec{F}$ is a conservative force. If yes, we are asked to find $U(x,y,z)$ and then find $\vec{F}$ back from $U$ and show it matches the original form.
Given $\vec{\nabla} \times\vec{F}=\vec{0}$, force is conservative.
Therefore, $U(x,y,z)=-\int_{r_0}^r \vec{F} \cdot d\vec{r}=-\int_0^x kydx-\int_0^y kxdy=-2xyk.$ (Note that the $z$ component is $0$).
Such a potential function yields $\vec{F}=-\vec{\nabla}U=2k\left<y,x,0\right>.$
Something must be wrong or I must be missing something because I get an extra factor of 2 and I do not understand why.