Answer
$f(x,y,z)=xe^{yz}+K$; Conservative
Work Step by Step
The vector field $F$ is conservative when $curl F=0$
When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$
Now, $curl F=[(xe^{yz}-xyze^{yz})-(xe^{yz}-xyze^{yz})]i+[(ye^{yz}-ye^{yz})]j+[(ze^{yz}-ze^{yz})-k=0$
Thus, we have the vector field $F$ is conservative.
Consider $f(x,y,z)=x(e^{yz})+g(y,z)$ and $f_y=xz(e^{yz})+g_y$
or, $g'(y)=0$
Thus, we have $g_y=h(z)$ Also, $f_y=xz(e^{yz}) \implies g(y,z)=y \sin z+h(z)$
Now, we have $f(x,y,z)=xe^{yz}+h(z) \implies f_z=xy(e^{yz})+h'(z)$
Thus, we get $h'(z)=0$
Hence, we have $f(x,y,z)=xe^{yz}+K$
