Answer
$ f_{xy}\neq f_{yx}$, then $f$ does not exist
Work Step by Step
Given $$\nabla f=\langle y^2,x\rangle$$
Since $$ \nabla f=\left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right\rangle=\langle y^2,x\rangle$$
Then
\begin{align*}
\frac{\partial f}{\partial x}&=y^2\ \ \ \Rightarrow\ \ f_{xy}=2y \\
\frac{\partial f}{\partial y}&=x\ \ \ \Rightarrow\ \ f_{yx}= 1 \
\end{align*}
Since $ f_{xy}\neq f_{yx}$, then $f$ does not exist
