The Value for Which Newton's Method Fails for the Function $y = - x ^ { 3 } + 3 x ^ { 2 } - x + 1$

Multiple Choice

The value for which Newton's method fails for the function below is shown in the graph. Give the reason why the method fails. $y = - x ^ { 3 } + 3 x ^ { 2 } - x + 1$

A) $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 0 = x _ { 1 } = x _ { 3 } = \ldots \\- 1 = x _ { 2 } = x _ { 4 } = \ldots\end{array} \right.$
B) $f ^ { \prime } \left( x _ { 1 } \right) = 0$
C) $f ^ { \prime } \left( x _ { 2 } \right) = 0$
D) $f ^ { \prime \prime } \left( x _ { 2 } \right) = 0$
E) $\lim _ { x \rightarrow \infty } \left\{ \begin{array} { l } 1 = x _ { 1 } = x _ { 3 } = \ldots \\0 = x _ { 2 } = x _ { 4 } = \ldots\end{array} \right.$