Answer
Conservative, $f(x,y,z)=xy^2z^3+k$
Work Step by Step
The vector field $F$ will be conservative if and only if $curl F=0$
consider $F=A i+B j+C k$
$curl F=[C_y-B_z]i+[A_z-C_z]j+[B_x-A_y]k$
Here, we have: $curl F=(6x yz^2 -6xyz^2)+(3y^2z^2-3y^2z^2)+(2yz^3-2yz^3)=0$
Thus, the vector field $F$ will be conservative. $f(x,y,z)=xy^2z^3+g(y,z)$
and $g'(y)=0$
Thus, $F_y=2xyz^3$
Now, $f(x,y,z)=xy^2z^3+h(z)$
and $h'(z)=0$
Thus, $F_z=3xy^2z^2$
Hence, $f(x,y,z)=xy^2z^3+k$
