A function $f ( x )$ defined for $x > a$ and a quartic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (Here, $a$ is a constant.) (a) For all real numbers $x > a$, $$( x - a ) f ( x ) = g ( x ).$$ (b) For two distinct real numbers $\alpha , \beta$, the function $f ( x )$ has the same local maximum value $M$ at $x = \alpha$ and $x = \beta$. (Here, $M > 0$) (c) The number of values of $x$ where the function $f ( x )$ has a local maximum or minimum is greater than the number of values of $x$ where the function $g ( x )$ has a local maximum or minimum. When $\beta - \alpha = 6 \sqrt { 3 }$, find the minimum value of $M$. [4 points]
A function $f ( x )$ defined for $x > a$ and a quartic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions.\\
(Here, $a$ is a constant.)\\
(a) For all real numbers $x > a$,
$$( x - a ) f ( x ) = g ( x ).$$
(b) For two distinct real numbers $\alpha , \beta$, the function $f ( x )$ has the same local maximum value $M$ at $x = \alpha$ and $x = \beta$. (Here, $M > 0$)\\
(c) The number of values of $x$ where the function $f ( x )$ has a local maximum or minimum is greater than the number of values of $x$ where the function $g ( x )$ has a local maximum or minimum.\\
When $\beta - \alpha = 6 \sqrt { 3 }$, find the minimum value of $M$. [4 points]