A real-valued function $f$ defined on a closed interval $[ a , b ]$ has the properties that $f ( a ) = f ( b ) = 0$ and $f ( x ) = f ^ { \prime } ( x ) + f ^ { \prime \prime } ( x )$ for all $x$ in $[ a , b ]$. Show that $f ( x ) = 0$ for all $x$ in $[ a , b ]$.