Problem $3$ Conservative forces and potential energy

The only force acting on a mass m in a $3-$dimensional space with coordinates x$,$ y$,$ and

z is given by the expression:

$$

F $=$ F$0($x$2$y$$x $+$ cx$3$y$)/$L$3$

In this expression, F$0=2\mathrm{.}00$N$,$ L $=0\mathrm{.}250$m$,$ and the value of the unknown dimension$-$

less constant c is to be determined in part a$.$ It is given that the force is conservative.

$($a$)$ Calculate the value of the constant c$.$

$($b$)$ In the $3-$dimensional space, a mass m $=0\mathrm{.}100$kg is constrained to follow a track

defined by the position vectors: