Answer
$U\left( x\right) =U_{0}+12x-3x^{2}=27+12x-3x^{2}$
Work Step by Step
Lets assume that our starting point is $x=0$ so the work done by conservative force at any given $x$ will be $W=\int ^{x}_{0}F\left( x\right) dx=\int ^{x}_{0}\left( 6x-12\right) =3x^{2}-12x(1)$ So the change in potential energy for conservative force will be: $\Delta U=-W\left( 2\right) $ So from (1) and (2) we get $\Delta U=-W=12x-3x^{2}(3)$ And also $\Delta U=U\left( x\right) -U_{0}\left( 4\right) $ From (3) and (4) we get: $U\left( x\right) =U_{0}+12x-3x^{2}$ in order $U(x)$ to be maximum positive $12x-3x^{2}$
