Question

Find the potential energy function V(x) for the following conservative force acting on an object of mass m, which starts from rest at x=0.

Fx=Fo+cx

Answer 1

The force on an object is the negative of the derivative of the potential function V (http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html#pec)

$F(x) = -\dfrac{dV}{dx}\,.$

$V(x) = - \int F\,dx = -\int (F_0+cx)\,dx = -F_0x - c\dfrac{x^2}{2} + \text{const}.$