Answer
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
Work Step by Step
$\textbf F(x,y) = y^2 z \textbf i + 2xyz \textbf j +xy^2 \textbf k$
$\nabla \times \textbf F =\begin{vmatrix}
\textbf i & \textbf j & \textbf k\\
\frac{\delta}{\delta x} & \frac{\delta}{\delta y}& \frac{\delta}{\delta z} \\
y^2 z & 2xyz & xy^2\\
\end{vmatrix} = (2xy - 2xy)\textbf i + (y^2-y^2)\textbf j + (2yz-2yz)\textbf k = 0
\\$
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
