Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f ( 0 ) = f ( 1 ) = 0$ and satisfies $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
If the function $\mathrm { e } ^ { - x } f ( x )$ assumes its minimum in the interval $[ 0,1 ]$ at $x = \frac { 1 } { 4 }$, which of the following is true?
(A) $f ^ { \prime } ( x ) < f ( x ) , \frac { 1 } { 4 } < x < \frac { 3 } { 4 }$
(B) $f ^ { \prime } ( x ) > f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(C) $f ^ { \prime } ( x ) < f ( x ) , \quad 0 < x < \frac { 1 } { 4 }$
(D) $f ^ { \prime } ( x ) < f ( x ) , \frac { 3 } { 4 } < x < 1$