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(C) Given the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$. Setting $x=0$ and $y=0$,we get $f(0) = f(0) + f(0) + 0 + 0$,which implies $f(0) = 0$. By

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$10$

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(C) Given the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$. Setting $x=0$ and $y=0$,we get $f(0) = f(0) + f(0) + 0 + 0$,which implies $f(0) = 0$. By the definition of the derivative,$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h}$. Using the given identity,$f(x+h) = f(x) + f(h) + xh^2 + x^2h$. Substituting this into the derivative formula: $f'(x) = \lim_{h \rightarrow 0} \frac{f(x) + f(h) + xh^2 + x^2h - f(x)}{h} = \lim_{h \rightarrow 0} \left( \frac{f(h)}{h} + xh + x^2 \right)$. Since $\lim_{h \rightarrow 0} \frac{f(h)}{h} = 1$ (given),we have $f'(x) = 1 + 0 + x^2 = 1 + x^2$. Therefore,$f'(3) = 1 + (3)^2 = 1 + 9 = 10$.

Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y) = f(x) + f(y) + xy^2 + x^2y$ for all real $x$ and $y$. If $\lim_{x \rightarrow 0} \frac{f(x)}{x} = 1$,then $f'(3)$ is equal to:

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