A function f(x) satisfies the relation `f(x+y) = f(x) + f(y) + xy(x+y), AA x, y in R`. If f'(0) = - 1, then

Options

A. f(x) is a polynomial function
B. f(x) is an exponential function
C. f(x) is twice differentiable for all `x in R`
D. f'(3) = 8

Answer

To solve the problem, we need to analyze the given functional equation and derive the properties of the function \( f(x) \).

### Step-by-step Solution:

1. **Understanding the Functional Equation**:
The given functional equation is:
\[ f(x+y) = f(x) + f(y) + xy(x+y) \]
for all \( x, y \in \mathbb{R} \).

2. **Finding \( f(0) \)**:
Let's substitute \( x = 0 \) and \( y = 0 \) into the functional equation:
\[ f(0+0) = f(0) + f(0) + 0 \cdot 0 \cdot (0+0) \]
This simplifies to:
\[ f(0) = 2f(0) \]
Rearranging gives:
\[ 0 = f(0) \]
Therefore, we find that:
\[ f(0) = 0 \]

3. **Differentiating the Functional Equation**:
To find \( f'(x) \), we differentiate both sides of the functional equation with respect to \( y \):
\[ \frac{d}{dy}[f(x+y)] = \frac{d}{dy}[f(x) + f(y) + xy(x+y)] \]
The left-hand side gives:
\[ f'(x+y) \]
The right-hand side gives:
\[ f'(y) + x(x+y) + xy \]
Thus, we have:
\[ f'(x+y) = f'(y) + x(x+y) + xy \]

4. **Setting \( y = 0 \)**:
Now, set \( y = 0 \):
\[ f'(x+0) = f'(0) + x(0) + 0 \]
This simplifies to:
\[ f'(x) = f'(0) \]
Since \( f'(0) = -1 \) (given), we have:
\[ f'(x) = -1 + x^2 \]

5. **Integrating to Find \( f(x) \)**:
Now, we integrate \( f'(x) \):
\[ f(x) = \int (-1 + x^2) \, dx = -x + \frac{x^3}{3} + C \]
To find \( C \), we use \( f(0) = 0 \):
\[ f(0) = -0 + \frac{0^3}{3} + C = 0 \implies C = 0 \]
Thus, we have:
\[ f(x) = -x + \frac{x^3}{3} \]

6. **Verifying the Properties**:
- **Polynomial Function**: The function \( f(x) = \frac{x^3}{3} - x \) is a polynomial function.
- **Twice Differentiable**: Since \( f(x) \) is a polynomial, it is differentiable infinitely many times.
- **Finding \( f'(3) \)**:
\[ f'(x) = -1 + x^2 \implies f'(3) = -1 + 3^2 = -1 + 9 = 8 \]

### Conclusion:
From the above steps, we conclude:
- \( f(x) \) is a polynomial function.
- \( f(x) \) is twice differentiable for all \( x \in \mathbb{R} \).
- \( f'(3) = 8 \).