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) $\frac { e ^ { 2 } - 4 } { 3 }$