Which quantity is obtained by taking the negative derivative of potential energy with respect to displacement in the case of a conservative force?

When a conservative force does positive work on a body,then the

$A$ body is moved along a closed loop. Is the work done in moving the body necessarily zero? If not,state the condition under which work done over a closed path is always zero.

Given below are two statements:
Statement $I$: An object moves from position $r_{1}$ to position $r_{2}$ under a conservative force field $\vec{F}$. The work done by the force is $W = -\int_{r_{1}}^{r_{2}} \vec{F} \cdot d\vec{r}$.
Statement $II$: Any object moving from one location to another location can follow an infinite number of paths. Therefore,the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements,choose the correct answer from the options given below:

In the case of non-conservative forces,the following statement is correct.

Conservative forces are defined as the force for which,