For a cubic function $f ( x )$ with leading coefficient $\frac { 1 } { 2 }$ and a real number $t$, let $g ( t )$ be the number of real roots of the equation $f ^ { \prime } ( x ) = 0$ in the closed interval $[ t , t + 2 ]$. The function $g ( t )$ satisfies the following conditions. (a) For all real numbers $a$, $\lim _ { t \rightarrow a + } g ( t ) + \lim _ { t \rightarrow a - } g ( t ) \leq 2$. (b) $g ( f ( 1 ) ) = g ( f ( 4 ) ) = 2 , g ( f ( 0 ) ) = 1$ Find the value of $f ( 5 )$. [4 points]
For a cubic function $f ( x )$ with leading coefficient $\frac { 1 } { 2 }$ and a real number $t$, let $g ( t )$ be the number of real roots of the equation $f ^ { \prime } ( x ) = 0$ in the closed interval $[ t , t + 2 ]$. The function $g ( t )$ satisfies the following conditions.\\
(a) For all real numbers $a$, $\lim _ { t \rightarrow a + } g ( t ) + \lim _ { t \rightarrow a - } g ( t ) \leq 2$.\\
(b) $g ( f ( 1 ) ) = g ( f ( 4 ) ) = 2 , g ( f ( 0 ) ) = 1$\\
Find the value of $f ( 5 )$. [4 points]