Two polynomial functions $f ( x )$ and $g ( x )$ satisfy
$$g ( x ) = \left( x ^ { 3 } + 2 \right) f ( x )$$
for all real numbers $x$. If $g ( x )$ has a local minimum value of 24 at $x = 1$, find the value of $f ( 1 ) - f ^ { \prime } ( 1 )$. [4 points]