The x-component of the force on a particle can be determined using the formula |F| = -dU/dx, where U represents potential energy. At x=15 cm, the force is zero, indicating a point of equilibrium. To find the force at x=5, 25, and 35 cm, one must calculate the slope of the potential energy graph at those points. A positive slope indicates increasing potential energy, while a negative slope indicates decreasing potential energy. Understanding these concepts is crucial for analyzing forces in one-dimensional motion.