Potential energy function along `x-`axis in a certain force field is given as
`U(x)=(x^(4))/(4)-2x^(2)+(11)/(2)x^(2)-6x`
For the given force field `:-`
(i)the points of equilibrium are `x=1`, `x=2` and `x=3`
(ii) the point `x=2` is a point of unstable equilibrium.
(iii) the points `x=1` and `x=3` are points of stable equilibrium.
(iv) there exists no point of neutral equilibrium. The correct option is `:-`

Options

A. `(i),(ii),(iv)`
B. `(i),(ii),(iii),(iv)`
C. (iii),(iv)
D. (ii),(iii)

Explanation

To solve the problem, we need to analyze the potential energy function given and determine the points of equilibrium, their stability, and the existence of neutral equilibrium.

### Step-by-Step Solution:

1. **Given Potential Energy Function**:
The potential energy function is given as:
\[ U(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x \]

2. **Finding Equilibrium Points**:
To find the points of equilibrium, we need to set the force equal to zero. The force \( F \) is related to the potential energy \( U \) by:
\[ F = -\frac{dU}{dx} \]
Therefore, we need to find \( \frac{dU}{dx} \) and set it to zero:
\[ \frac{dU}{dx} = x^3 - 6x^2 + 11x - 6 \]
Setting this equal to zero:
\[ x^3 - 6x^2 + 11x - 6 = 0 \]
By solving this cubic equation, we find the roots, which are:
\[ x = 1, \quad x = 2, \quad x = 3 \]
Thus, the points of equilibrium are \( x = 1, 2, 3 \).

3. **Stability Analysis**:
To determine the stability of these equilibrium points, we need to compute the second derivative of the potential energy function:
\[ \frac{d^2U}{dx^2} = 3x^2 - 12x + 11 \]
We will evaluate this second derivative at each of the equilibrium points.

- **At \( x = 1 \)**:
\[ \frac{d^2U}{dx^2} \bigg|_{x=1} = 3(1)^2 - 12(1) + 11 = 3 - 12 + 11 = 2 \quad (> 0) \]
This indicates that \( x = 1 \) is a point of stable equilibrium.

- **At \( x = 2 \)**:
\[ \frac{d^2U}{dx^2} \bigg|_{x=2} = 3(2)^2 - 12(2) + 11 = 12 - 24 + 11 = -1 \quad (< 0) \]
This indicates that \( x = 2 \) is a point of unstable equilibrium.

- **At \( x = 3 \)**:
\[ \frac{d^2U}{dx^2} \bigg|_{x=3} = 3(3)^2 - 12(3) + 11 = 27 - 36 + 11 = 2 \quad (> 0) \]
This indicates that \( x = 3 \) is a point of stable equilibrium.

4. **Neutral Equilibrium**:
We check if there exists any point where \( \frac{d^2U}{dx^2} = 0 \):
\[ 3x^2 - 12x + 11 = 0 \]
The discriminant of this quadratic equation is:
\[ D = (-12)^2 - 4 \cdot 3 \cdot 11 = 144 - 132 = 12 \quad (> 0) \]
This means there are two real roots, but since we are looking for points of neutral equilibrium, we need to check if these roots correspond to equilibrium points. However, since we already found the equilibrium points and they do not satisfy \( \frac{d^2U}{dx^2} = 0 \), we conclude that there are no points of neutral equilibrium.

### Conclusion:
The analysis yields the following results:
- The points of equilibrium are \( x = 1, 2, 3 \).
- \( x = 1 \) and \( x = 3 \) are points of stable equilibrium.
- \( x = 2 \) is a point of unstable equilibrium.
- There exists no point of neutral equilibrium.

Thus, the correct option is that all statements about the equilibrium points are correct.