Answer
Plot $I$
Work Step by Step
The level sets of $f$ are circles, so the vector filed must be perpendicular to circles, leaving just $I$ and $III$ as possible plots. Noting that the vectors of the gradient vector filed, $\frac{cos(x^{2}+y^{2})}{\sqrt {x^{2}+y^{2}}}\lt x,y\gt$ , it can be seen that the field point towards the origin as well as away.
Hence, the correct plot must be $I$.
