Question:

Step 1: Determine $f(0)$ and interpret the given limit to find $f'(0)$. First, we use the given functional equation to determine the value of $f(0)$. Then, we interpret the given limit condition in terms of the derivative at $x=0$. From the functional equation $f(x+y) = f(x) + f(y) + x^2y + xy^2$, setting $x=0$ and $y=0$ gives: $$f(0+0) = f(0) + f(0) + 0^2 \cdot 0 + 0 \cdot 0^2$$ $$f(0) = f(0) + f(0)$$ This implies $f(0) = 0$. Now, consider the definition of the derivative at $x=0$: $$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}$$ Since $f(0)=0$, this becomes: $$f'(0) = \lim_{h \to 0} \frac{f(h)}{h}$$ We are given that $ \lim_{x \to 0}\left[\dfrac{f(x)}{x}\right] = 1 $. Therefore, $f'(0) = 1$. Step 2: Differentiate the functional equation to find $f'(x)$. Next, we differentiate the given functional equation with respect to $y$, treating $x$ as a constant, to find a relationship involving derivatives. Then, we substitute $y=0$ and the value of $f'(0)$ found in the previous step to determine the general form of $f'(x)$. The given functional equation is: $$f(x+y) = f(x) + f(y) + x^2y + xy^2$$ Differentiating both sides with respect to $y$ (keeping $x$ constant), we apply the chain rule on the left side: $$\frac{d}{dy}[f(x+y)] = \frac{d}{dy}[f(x)] + \frac{d}{dy}[f(y)] + \frac{d}{dy}[x^2y] + \frac{d}{dy}[xy^2]$$ $$f'(x+y) \cdot \frac{d}{dy}(x+y) = 0 + f'(y) + x^2 \cdot 1 + x \cdot 2y$$ $$f'(x+y) = f'(y) + x^2 + 2xy$$ Now, substitute $y=0$ into this equation: $$f'(x+0) = f'(0) + x^2 + 2x(0)$$ $$f'(x) = f'(0) + x^2$$ Using $f'(0) = 1$ from Step 1: $$f'(x) = 1 + x^2$$ Step 3: Integrate $f'(x)$ to find $f(x)$. With the expression for $f'(x)$, we integrate it to find $f(x)$. We use the value of $f(0)$ previously determined to find the constant of integration. We have $f'(x) = 1 + x^2$. Integrating $f'(x)$ with respect to $x$ gives $f(x)$: $$f(x) = \int (1 + x^2) dx$$ $$f(x) = x + \frac{x^3}{3} + C$$ To find the constant of integration $C$, we use the condition $f(0)=0$: $$f(0) = 0 + \frac{0^3}{3} + C$$ $$0 = C$$ Thus, the function $f(x)$ is: $$f(x) = x + \frac{x^3}{3}$$ Step 4: Calculate $f(3)$. Finally, we substitute $x=3$ into the derived expression for $f(x)$ to find the required value. Substitute $x=3$ into $f(x) = x + \frac{x^3}{3}$: $$f(3) = 3 + \frac{3^3}{3}$$ $$f(3) = 3 + \frac{27}{3}$$ $$f(3) = 3 + 9$$ $$f(3) = 12$$ The value of $f(3)$ is $12$.