A cubic function $f ( x )$ with leading coefficient 1 and a quadratic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (가) The tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = g ( x )$ at the point $( 2,0 )$ are both the $x$-axis. (나) The number of tangent lines to the curve $y = f ( x )$ drawn from the point $( 2,0 )$ is 2. (다) The equation $f ( x ) = g ( x )$ has exactly one real root. For all real numbers $x > 0$, $$g ( x ) \leq k x - 2 \leq f ( x )$$ Let $\alpha$ and $\beta$ be the maximum and minimum values of the real number $k$ satisfying the above inequality, respectively. When $\alpha - \beta = a + b \sqrt { 2 }$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$ and $b$ are rational numbers.) [4 points]
A cubic function $f ( x )$ with leading coefficient 1 and a quadratic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions.\\
(가) The tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = g ( x )$ at the point $( 2,0 )$ are both the $x$-axis.\\
(나) The number of tangent lines to the curve $y = f ( x )$ drawn from the point $( 2,0 )$ is 2.\\
(다) The equation $f ( x ) = g ( x )$ has exactly one real root.\\
For all real numbers $x > 0$,
$$g ( x ) \leq k x - 2 \leq f ( x )$$
Let $\alpha$ and $\beta$ be the maximum and minimum values of the real number $k$ satisfying the above inequality, respectively. When $\alpha - \beta = a + b \sqrt { 2 }$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$ and $b$ are rational numbers.) [4 points]