question_answer

Let\[f\,\,(x)=\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{n}^{n}}(x+n)\,\,\left( x\,\,\frac{n}{2} \right)....\left( x\,\,\frac{n}{n} \right)}{n!({{x}^{2}}+{{n}^{2}})\,\,\left( {{x}^{2}}+\frac{{{n}^{4}}}{4} \right)....\left( {{x}^{2}}+\frac{{{n}^{2}}}{{{n}^{2}}} \right)} \right)}^{\frac{x}{n}}},\]for all x >0. Then

A)  \[f\left( \frac{1}{2} \right)\,\underline{<}\,\,f(1)\]              

B)  \[f\left( \frac{1}{3} \right)\,\underline{<}\,\,f\left( \frac{2}{3} \right)\]

C)  \[f'(2)\,\underline{>}\,\,0\]                     

D)  Both (a) and (b)

E)  All of these