For $a > 1$, consider the function $f ( x ) = 2 x ^ { 3 } - 3 ( a + 1 ) x ^ { 2 } + 6 a x - 4 a + 2$. Let $b$ be one real root of the equation $f ( x ) = 0$. The following is a process for comparing the magnitudes of two numbers $a$ and $b$. $f ^ { \prime } ( x ) =$ (가) and since $a > 1$, $f ( x )$ has a (나) at $x = 1$. Since $f ( 1 ) < 0$ and $f ( b ) = 0$, $a$ (다) $b$.
What are the correct expressions for (가), (나), and (다) in the above process? [3 points]
| (가) | (나) | (다) |
| (1) $6 ( x - 1 ) ( x - a )$ | local minimum | $>$ |
| (2) $6 ( x - 1 ) ( x - a )$ | local minimum | $<$ |
| (3) $6 ( x - 1 ) ( x - a )$ | local maximum | $>$ |
| (4) $6 ( x - a ) ( x - 1 )$ | local maximum | $<$ |
| (5) $6 ( x - a ) ( x - 1 )$ | local maximum | $>$ |