For two natural numbers $a$ and $b$, the function $f(x)$ is defined as $$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$ For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$. $$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$ If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points] (1) 51 (2) 52 (3) 53 (4) 54 (5) 55
For two natural numbers $a$ and $b$, the function $f(x)$ is defined as
$$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$
For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$.
$$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$
If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points]\\
(1) 51\\
(2) 52\\
(3) 53\\
(4) 54\\
(5) 55