[tex]\dfrac{\partial f}{\partial x}=3y^2z^3\implies f(x,y,z)=3xy^2z^3+g(y,z)[/tex]
[tex]\dfrac{\partial f}{\partial y}=6xyz^3+\dfrac{\partial g}{\partial y}=6xyz^3[/tex]
[tex]\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)[/tex]
[tex]\dfrac{\partial f}{\partial z}=9xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}=9xy^2z^2[/tex]
[tex]\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C[/tex]
[tex]\implies f(x,y,z)=3xy^2z^3+C[/tex]
The potential function exists, so [tex]\mathbf f[/tex] is indeed conservative.
