Answer
The vector field $F$ is irrotational.
Work Step by Step
The vector field $F$ will be irrotational and conservative when $curl F=0$
When $F=ai+bj+ck$, then we have $curl F=[c_y-b_z]i+[a_z-c_z]j+[b_x-a_y]k$
We are given that $F(x,y,z)=f(x) i+g(y) j+h(z) k$
Here, we have $curl F= \nabla \times F=[\dfrac{\partial h(z)}{\partial y}-\dfrac{\partial g(y)}{\partial z}]i+[\dfrac{\partial f(x)}{\partial z}-\dfrac{\partial h(z)}{\partial x}]j+[\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y}]k=0$
Hence, the vector field $F$ is irrotational.
