Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options. (1) $1 - 2i$ is also a root of $f ( x ) = 0$ (2) Both $a$ and $b$ are positive numbers (3) $f ^ { \prime } ( 2.1 ) < 0$ (4) The function $y = f ( x )$ has a local minimum at $x = 1$ (5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0
Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options.\\
(1) $1 - 2i$ is also a root of $f ( x ) = 0$\\
(2) Both $a$ and $b$ are positive numbers\\
(3) $f ^ { \prime } ( 2.1 ) < 0$\\
(4) The function $y = f ( x )$ has a local minimum at $x = 1$\\
(5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0