A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The function $f ( x )$ has a local maximum at $x = 0$ and a local minimum at $x = k$. (Here, $k$ is a constant.) (나) For all real numbers $t$ greater than 1, $\int _ { 0 } ^ { t } \left| f ^ { \prime } ( x ) \right| d x = f ( t ) + f ( 0 )$ Which of the following statements in the given options are correct? [4 points] Options ᄀ. $\int _ { 0 } ^ { k } f ^ { \prime } ( x ) d x < 0$ ㄴ. $0 < k \leq 1$ ㄷ. The local minimum value of the function $f ( x )$ is 0. (1) ᄀ (2) ㄷ (3) ᄀ, ㄴ (4) ㄴ, ㄷ (5) ᄀ, ㄴ, ㄷ
A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions.\\
(가) The function $f ( x )$ has a local maximum at $x = 0$ and a local minimum at $x = k$. (Here, $k$ is a constant.)\\
(나) For all real numbers $t$ greater than 1,\\
$\int _ { 0 } ^ { t } \left| f ^ { \prime } ( x ) \right| d x = f ( t ) + f ( 0 )$\\
\\
Which of the following statements in the given options are correct? [4 points]\\
Options\\
ᄀ. $\int _ { 0 } ^ { k } f ^ { \prime } ( x ) d x < 0$\\
ㄴ. $0 < k \leq 1$\\
ㄷ. The local minimum value of the function $f ( x )$ is 0.\\
(1) ᄀ\\
(2) ㄷ\\
(3) ᄀ, ㄴ\\
(4) ㄴ, ㄷ\\
(5) ᄀ, ㄴ, ㄷ