Potential Energy Equation:
The potential energy of a body is given by the equation A - Bx^2, where x is the displacement.

Explanation:
To determine the force acting on the particle, we need to differentiate the potential energy equation with respect to displacement (x).

Differentiation:
Differentiating the given equation, we get:
dU/dx = -2Bx

Interpretation:
The above expression represents the rate of change of potential energy with respect to displacement. This is equivalent to the force acting on the particle.

Magnitude of Force:
From the differentiation, we can observe that the magnitude of force acting on the particle is proportional to the displacement (x).

Answer:
Therefore, the correct answer is option B: Proportional to x.

Explanation in Detail:
The potential energy of a body is a scalar quantity that depends on the position or configuration of the body. It is defined as the work done by an external force to bring the body from a reference position to its current position. In this case, the potential energy equation is given as A - Bx^2, where A and B are constants.

To determine the force acting on the particle, we can differentiate the potential energy equation with respect to displacement (x). This will give us the rate of change of the potential energy with respect to displacement, which is equivalent to the force acting on the particle.

By differentiating the given equation, we get:

dU/dx = -2Bx

Here, dU/dx represents the rate of change of potential energy (U) with respect to displacement (x). The negative sign indicates that the force is in the opposite direction of the displacement.

From the expression, we can observe that the magnitude of the force acting on the particle is proportional to the displacement (x). As the displacement increases, the force also increases proportionally. Similarly, if the displacement decreases, the force decreases proportionally.

Hence, the correct answer is option B: Proportional to x.

F= -dU/dx. after derivation F= 2Bx.