For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points] $\langle$Options$\rangle$ ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$. ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$. ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then $$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$ (1) ㄱ (2) ㄴ (3) ㄱ, ㄴ (4) ㄱ, ㄷ (5) ㄱ, ㄴ, ㄷ
For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points]
$\langle$Options$\rangle$
\noindent ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$.\\
ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$.\\
ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then
$$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$
(1) ㄱ\\
(2) ㄴ\\
(3) ㄱ, ㄴ\\
(4) ㄱ, ㄷ\\
(5) ㄱ, ㄴ, ㄷ