A particle oscillates about the equilibrium position x₀ subject to a force that has an associated potential energy U(x). Which of the following statements (perhaps more than one ) about U(x) is true?

(A) U(x) must be symmetric about x₀.

(B) U(x) must have a minimum at x₀.

(C) U(x) may have a maximum at x₀.

(D) U(x) must be positive in the vicinity of x₀.

The correct option is (A), (B) and (D).
Since the particle oscillates about the equilibrium position x₀, therefore the potential energy U(x) must be symmetric about x₀. Thus option (A) is correct.
The force and potential energy is related by F_x(x) = -dU/dx. The force must always act to restore the particle to equilibrium, and the potential energy must have a minimum at the equilibrium position. Thus option (B) is correct.
Since the particle oscillates about the equilibrium position x₀, so the potential energy U(x) must be symmetric about x₀ and the potential energy must have a minimum value at the equilibrium position. Therefore U(x) must be positive in the vicinity of x₀. Thus option (D) is correct.