JEE Math Practice Question
Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(1)=\frac{1}{3}$ and $3 \int_{1}^{x} f(t) d t=x f(x)-\frac{x^{3}}{3}, x \in[1, \infty)$. Let $e$ denote the base of the natural logarithm. Then the value of $f(e)$ is
- A.$\frac{e^{2}+4}{3}$
- B.$\frac{\log _{e} 4+e}{3}$
- C.$\frac{4 e^{2}}{3}$
- D.$\frac{e^{2}-4}{3}$
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