A particle free to move along x-axis is acted upon by a force `F=-ax+bx^(2) whrte a and b are positive constants. `, the correct variation of potential energy function U(x) is best represented by.

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to find the potential energy function \( U(x) \) corresponding to the given force \( F(x) = -ax + bx^2 \). The relationship between force and potential energy is given by:

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

### Step 1: Write the expression for potential energy
From the relationship above, we can express the change in potential energy as:

\[ dU = -F \, dx \]

Substituting the expression for \( F \):

\[ dU = -(-ax + bx^2) \, dx = (ax - bx^2) \, dx \]

### Step 2: Integrate to find \( U(x) \)
Now, we integrate to find the total potential energy \( U(x) \):

\[ U(x) = \int (ax - bx^2) \, dx \]

Calculating the integral:

\[ U(x) = \int ax \, dx - \int bx^2 \, dx \]

\[ U(x) = \frac{a}{2} x^2 - \frac{b}{3} x^3 + C \]

Where \( C \) is the constant of integration.

### Step 3: Determine the points where \( U(x) = 0 \)
To find the points where the potential energy is zero, we set \( U(x) = 0 \):

\[ \frac{a}{2} x^2 - \frac{b}{3} x^3 + C = 0 \]

Assuming \( C = 0 \) for simplicity (we can choose the reference point for potential energy), we have:

\[ \frac{a}{2} x^2 - \frac{b}{3} x^3 = 0 \]

Factoring out \( x^2 \):

\[ x^2 \left( \frac{a}{2} - \frac{b}{3} x \right) = 0 \]

This gives us two solutions:

1. \( x = 0 \)
2. \( \frac{a}{2} - \frac{b}{3} x = 0 \) which leads to \( x = \frac{3a}{2b} \)

### Step 4: Analyze the behavior of \( U(x) \)
- At \( x = 0 \), \( U(0) = 0 \).
- As \( x \) increases from 0, the term \( \frac{a}{2} x^2 \) dominates initially, making \( U(x) \) positive.
- As \( x \) continues to increase, the \( -\frac{b}{3} x^3 \) term will eventually dominate, causing \( U(x) \) to decrease and become negative after reaching a maximum.

### Step 5: Determine the correct graph
Given the behavior of \( U(x) \):
- It starts at 0 when \( x = 0 \).
- It increases to a maximum and then decreases, eventually becoming negative.

Thus, the correct representation of the potential energy function \( U(x) \) is best represented by option C, which shows the potential energy starting at zero, increasing, reaching a maximum, and then decreasing through zero into negative values.

### Summary of Steps:
1. Write the expression for potential energy using the force.
2. Integrate to find the potential energy function.
3. Set the potential energy function to zero to find critical points.
4. Analyze the behavior of the potential energy function.
5. Identify the correct graph representation.