Consider the function $f ( x ) = x ^ { n } ( 1 - x ) ^ { n } / n !$, where $n \geq 1$ is a fixed integer. Let $f ^ { ( k ) }$ denote the $k$-th derivative of $f$. Which of the following is true for all $k \geq 1$? (a) $f ^ { ( k ) } ( 0 )$ and $f ^ { ( k ) } ( 1 )$ are integers. (b) $f ^ { ( k ) } ( 0 )$ is an integer, but not $f ^ { ( k ) } ( 1 )$ (c) $f ^ { ( k ) } ( 1 )$ is an integer, but not $f ^ { ( k ) } ( 0 )$ (d) Neither $f ^ { ( k ) } ( 1 )$ nor $f ^ { ( k ) } ( 0 )$ is an integer.
(A) For $n=3$: $f^{(3)}(0) = 1$ and $f^{(3)}(1) = -1$, both integers. In general $f^{(k)}(x)$ always yields integers at $x=0$ and $x=1$.
Consider the function $f ( x ) = x ^ { n } ( 1 - x ) ^ { n } / n !$, where $n \geq 1$ is a fixed integer. Let $f ^ { ( k ) }$ denote the $k$-th derivative of $f$. Which of the following is true for all $k \geq 1$?\\
(a) $f ^ { ( k ) } ( 0 )$ and $f ^ { ( k ) } ( 1 )$ are integers.\\
(b) $f ^ { ( k ) } ( 0 )$ is an integer, but not $f ^ { ( k ) } ( 1 )$\\
(c) $f ^ { ( k ) } ( 1 )$ is an integer, but not $f ^ { ( k ) } ( 0 )$\\
(d) Neither $f ^ { ( k ) } ( 1 )$ nor $f ^ { ( k ) } ( 0 )$ is an integer.