The function $f$ has derivatives of all orders for all real numbers. It is known that $\left| f ^ { ( 4 ) } ( x ) \right| \leq \frac { 12 } { 5 }$ and $\left| f ^ { ( 5 ) } ( x ) \right| \leq \frac { 3 } { 2 }$ for $0 \leq x \leq 2$. Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. The Taylor series for $f$ about $x = 0$ converges at $x = 2$. Of the following, which is the smallest value of $k$ for which the Lagrange error bound guarantees that $\left| f ( 2 ) - P _ { 4 } ( 2 ) \right| \leq k$? (A) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 3 } { 2 }$ (B) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 12 } { 5 }$ (C) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 3 } { 2 }$ (D) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 12 } { 5 }$
The function $f$ has derivatives of all orders for all real numbers. It is known that $\left| f ^ { ( 4 ) } ( x ) \right| \leq \frac { 12 } { 5 }$ and $\left| f ^ { ( 5 ) } ( x ) \right| \leq \frac { 3 } { 2 }$ for $0 \leq x \leq 2$. Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. The Taylor series for $f$ about $x = 0$ converges at $x = 2$. Of the following, which is the smallest value of $k$ for which the Lagrange error bound guarantees that $\left| f ( 2 ) - P _ { 4 } ( 2 ) \right| \leq k$?\\
(A) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 3 } { 2 }$\\
(B) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 12 } { 5 }$\\
(C) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 3 } { 2 }$\\
(D) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 12 } { 5 }$