Let $A = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { c c } 1 , & \text { if } i = j \\ - x , & \text { if } | i - j | = 1 \\ 2 x + 1 , & \text { otherwise } \end{array} \right.$ Let a function $f : R \rightarrow R$ be defined as $f ( x ) = \operatorname { det } ( A )$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to: (1) $- \frac { 20 } { 27 }$ (2) $\frac { 88 } { 27 }$ (3) $\frac { 20 } { 27 }$ (4) $- \frac { 88 } { 27 }$
Let $A = \left[ a _ { i j } \right]$ be a $3 \times 3$ matrix, where $a _ { i j } = \left\{ \begin{array} { c c } 1 , & \text { if } i = j \\ - x , & \text { if } | i - j | = 1 \\ 2 x + 1 , & \text { otherwise } \end{array} \right.$\\
Let a function $f : R \rightarrow R$ be defined as $f ( x ) = \operatorname { det } ( A )$. Then the sum of maximum and minimum values of $f$ on $R$ is equal to:\\
(1) $- \frac { 20 } { 27 }$\\
(2) $\frac { 88 } { 27 }$\\
(3) $\frac { 20 } { 27 }$\\
(4) $- \frac { 88 } { 27 }$