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 $f ^ { \prime \prime } ( x ) - 2 f ^ { \prime } ( x ) + f ( x ) \geq e ^ { x } , x \in [ 0,1 ]$.
Which of the following is true for $0 < x < 1$?
(A) $0 < f ( x ) < \infty$
(B) $- \frac { 1 } { 2 } < f ( x ) < \frac { 1 } { 2 }$
(C) $- \frac { 1 } { 4 } < f ( x ) < 1$
(D) $- \infty < f ( x ) < 0$