For a real number $t$, define the function $f ( x )$ as $$f ( x ) = \left\{ \begin{array} { c c }
1 - | x - t | & ( | x - t | \leq 1 ) \\
0 & ( | x - t | > 1 )
\end{array} \right.$$ For a certain odd number $k$, the function $$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$ satisfies the following condition. When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$. Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]
For a real number $t$, define the function $f ( x )$ as
$$f ( x ) = \left\{ \begin{array} { c c }
1 - | x - t | & ( | x - t | \leq 1 ) \\
0 & ( | x - t | > 1 )
\end{array} \right.$$
For a certain odd number $k$, the function
$$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$
satisfies the following condition.
When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$.\\
Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]