Let f:R→R be a differentiable function satisfying f(x+y)=f(x)+f(y)+x2y+xy2 for all real number x and y. If limx→0 f(x)x=1,then 

Solution:

Given that \( f: \mathbb{R} \to \mathbb{R} \) is a differentiable function satisfying \( f(x + y) = f(x) + f(y) + x^2 y + xy^2 \) for all real numbers \( x \) and \( y \), and \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \), we need to find \( f(x) \).

Step-by-Step Solution:

1. Using the Functional Equation:

   Let \( y = 0 \):

   \( f(x + 0) = f(x) + f(0) + x^2 \cdot 0 + x \cdot 0^2 \)

   \( f(x) = f(x) + f(0) \)

   \( f(0) = 0 \)

2. Differentiate Both Sides with Respect to \( y \):

   Differentiate the original equation with respect to \( y \):

   \( \frac{\partial}{\partial y} f(x + y) = \frac{\partial}{\partial y} [f(x) + f(y) + x^2 y + xy^2] \)

   \( f'(x + y) = f'(y) + x^2 + 2xy \)

3. Evaluate at \( y = 0 \):

   \( f'(x) = f'(0) + x^2 \)

4. Given \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \):

   This implies that the derivative of \( f(x) \) at \( x = 0 \) is 1:

   \( f'(0) = 1 \)

5. Combine the Results:

   \( f'(x) = 1 + x^2 \)

6. Integrate to Find \( f(x) \):

   Integrate \( f'(x) \) with respect to \( x \):

   \( f(x) = \int (1 + x^2) \, dx \)

   \( f(x) = x + \frac{x^3}{3} + C \)

   Since \( f(0) = 0 \), \( C = 0 \):

   \( f(x) = x + \frac{x^3}{3} \)