question_answer

Let f (x) be differentiable on the interval \[(0,\infty )\] such that \[f(1)=1,\] and \[\underset{t\to x}{\mathop{\lim }}\,\frac{{{t}^{2}}f\,\,(x)-{{x}^{2}}f\,\,(t)}{t-x}=1\] for each x > 0. Then f (x) is:

A)  \[\frac{1}{3x}+\frac{2{{x}^{2}}}{3}\]

B)  \[-\frac{1}{3x}+\frac{4{{x}^{2}}}{3}\]

C)  \[-\frac{1}{x}+\frac{2}{{{x}^{2}}}\] 

D)  \[\frac{1}{x}\] 

E)  None of these