Final answer:
OPTION B.A particle in the potential function U(x) = a/x + bx is in equilibrium at the point where the force is zero, which corresponds to the minimum of the potential energy function, signifying a stable equilibrium.
Step-by-step explanation:
The particle is in equilibrium when the force acting on it is zero. The force can be obtained by taking the negative derivative of the potential energy function U(x). So, if U(x) = a/x + bx, then the force F(x) = -dU/dx. The equilibrium points occur where this force is equal to zero, which implies that the potential energy function has a stationary point at these locations. Since a and b are positive constants, the function has two components, one that decreases as x increases (a/x) and one that increases as x increases (bx).
By finding where the derivative of U(x) is zero, one can figure out where the potential energy has either a minimum or a maximum. In this case, at the point of minimum potential energy, the particle will be in a stable equilibrium because small displacements from this point will result in forces that push the particle back towards the equilibrium. Conversely, at a point of maximum potential energy, the equilibrium would be unstable as any small displacement will result in forces pushing the particle further away. Hence, a particle in a potential function such as U(x) = a/x + bx, where a and b are positive, will be in equilibrium at the point with minimum potential energy.