Suppose that \( f \) is a differentiable function with the property that \( f(x+y)=f(x)+f(y)+x y \) and \( \lim _{h \rightarrow 0} \frac{1}{h} f(h)=3 \) where [.] represents greatest integer function, then
\( \mathrm{P} \)
(A) \( f \) is a linear function
(B) \( 2 f(1)=\left[\lim _{x \rightarrow 0}(1+2 x)^{1 / x}\right] \)
\( (C) f(x)=3 x+\frac{x^{2}}{2} \)
(D) \( f^{\prime}(1)=4 \)
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