A particle of mass m moves in a force field such that its potential energy in force field is defined by the equation `U = +A (x-a)^(2) (x-b)^(2)`. Where A, a and b are `+ve` constants then body may oscillate simple harmonically about point.

Options

A. a only
B. a and b both
C. only `(a+b)/(2)`
D. b only

Explanation

To determine the points about which the particle may oscillate simple harmonically, we need to analyze the potential energy function given by:

\[ U = A (x - a)^2 (x - b)^2 \]

### Step 1: Find the force from the potential energy
The force \( F \) acting on the particle can be derived from the potential energy \( U \) using the relation:

\[ F = -\frac{dU}{dx} \]

### Step 2: Differentiate the potential energy function
We need to differentiate \( U \) with respect to \( x \):

1. First, apply the product rule since \( U \) is a product of two functions:
\[ U = A (x - a)^2 (x - b)^2 \]

2. Let \( f(x) = (x - a)^2 \) and \( g(x) = (x - b)^2 \).
Then, using the product rule:
\[ \frac{dU}{dx} = A \left( f'(x)g(x) + f(x)g'(x) \right) \]

3. Calculate \( f'(x) \) and \( g'(x) \):
\[ f'(x) = 2(x - a) \]
\[ g'(x) = 2(x - b) \]

4. Substitute back into the derivative:
\[ \frac{dU}{dx} = A \left( 2(x - a)(x - b)^2 + (x - a)^2 \cdot 2(x - b) \right) \]
\[ = 2A \left( (x - a)(x - b)^2 + (x - a)^2(x - b) \right) \]

### Step 3: Set the force to zero for equilibrium points
To find the equilibrium points, set \( F = -\frac{dU}{dx} = 0 \):

\[ 2A \left( (x - a)(x - b)^2 + (x - a)^2(x - b) \right) = 0 \]

This implies:

\[ (x - a)(x - b)^2 + (x - a)^2(x - b) = 0 \]

### Step 4: Factor the equation
Factoring out the common terms:

\[ (x - a)(x - b) \left( (x - b) + (x - a) \right) = 0 \]

This gives us the solutions:

1. \( x - a = 0 \) → \( x = a \)
2. \( x - b = 0 \) → \( x = b \)
3. \( (x - b) + (x - a) = 0 \) → \( 2x - (a + b) = 0 \) → \( x = \frac{a + b}{2} \)

### Step 5: Determine stability of equilibrium points
To determine if these points are stable equilibrium points, we need to check the second derivative of \( U \):

1. Calculate \( \frac{d^2U}{dx^2} \).
2. Evaluate \( \frac{d^2U}{dx^2} \) at \( x = a \) and \( x = b \).

If \( \frac{d^2U}{dx^2} > 0 \), the equilibrium is stable.

### Conclusion
The points about which the particle may oscillate simple harmonically are \( x = a \) and \( x = b \).