The function $f ( x )$ is $$f ( x ) = \begin{cases} - x ^ { 2 } & ( x < 0 ) \\ x ^ { 2 } - x & ( x \geq 0 ) \end{cases}$$ and for a positive number $a$, the function $g ( x )$ is $$g ( x ) = \left\{ \begin{array} { c l }
a x + a & ( x < - 1 ) \\
0 & ( - 1 \leq x < 1 ) \\
a x - a & ( x \geq 1 )
\end{array} \right.$$ Let $k$ be the maximum value of $a$ such that the function $h ( x ) = \int _ { 0 } ^ { x } ( g ( t ) - f ( t ) ) d t$ has exactly one extremum. When $a = k$, what is the value of $k + h ( 3 )$? [4 points] (1) $\frac { 9 } { 2 }$ (2) $\frac { 11 } { 2 }$ (3) $\frac { 13 } { 2 }$ (4) $\frac { 15 } { 2 }$ (5) $\frac { 17 } { 2 }$
The function $f ( x )$ is
$$f ( x ) = \begin{cases} - x ^ { 2 } & ( x < 0 ) \\ x ^ { 2 } - x & ( x \geq 0 ) \end{cases}$$
and for a positive number $a$, the function $g ( x )$ is
$$g ( x ) = \left\{ \begin{array} { c l }
a x + a & ( x < - 1 ) \\
0 & ( - 1 \leq x < 1 ) \\
a x - a & ( x \geq 1 )
\end{array} \right.$$
Let $k$ be the maximum value of $a$ such that the function $h ( x ) = \int _ { 0 } ^ { x } ( g ( t ) - f ( t ) ) d t$ has exactly one extremum. When $a = k$, what is the value of $k + h ( 3 )$? [4 points]\\
(1) $\frac { 9 } { 2 }$\\
(2) $\frac { 11 } { 2 }$\\
(3) $\frac { 13 } { 2 }$\\
(4) $\frac { 15 } { 2 }$\\
(5) $\frac { 17 } { 2 }$