The position $x ( t )$ of a point P moving on a number line at time $t$ is given by $$x ( t ) = t ( t - 1 ) ( a t + b ) \quad ( a \neq 0 )$$ for two constants $a , b$. The velocity $v ( t )$ of point P at time $t$ satisfies $\int _ { 0 } ^ { 1 } | v ( t ) | d t = 2$. Which of the following statements in the given options are correct? [4 points] Given statements: ᄀ. $\int _ { 0 } ^ { 1 } v ( t ) d t = 0$ ㄴ. There exists $t _ { 1 }$ in the open interval $( 0,1 )$ such that $\left| x \left( t _ { 1 } \right) \right| > 1$. ㄷ. If $| x ( t ) | < 1$ for all $t$ with $0 \leq t \leq 1$, then there exists $t _ { 2 }$ in the open interval $( 0,1 )$ such that $x \left( t _ { 2 } \right) = 0$. (1) ᄀ (2) ᄀ, ㄴ (3) ᄀ, ㄷ (4) ㄴ, ㄷ (5) ᄀ, ㄴ, ㄷ
The position $x ( t )$ of a point P moving on a number line at time $t$ is given by
$$x ( t ) = t ( t - 1 ) ( a t + b ) \quad ( a \neq 0 )$$
for two constants $a , b$. The velocity $v ( t )$ of point P at time $t$ satisfies $\int _ { 0 } ^ { 1 } | v ( t ) | d t = 2$. Which of the following statements in the given options are correct? [4 points]
Given statements:\\
ᄀ. $\int _ { 0 } ^ { 1 } v ( t ) d t = 0$\\
ㄴ. There exists $t _ { 1 }$ in the open interval $( 0,1 )$ such that $\left| x \left( t _ { 1 } \right) \right| > 1$.\\
ㄷ. If $| x ( t ) | < 1$ for all $t$ with $0 \leq t \leq 1$, then there exists $t _ { 2 }$ in the open interval $( 0,1 )$ such that $x \left( t _ { 2 } \right) = 0$.\\
(1) ᄀ\\
(2) ᄀ, ㄴ\\
(3) ᄀ, ㄷ\\
(4) ㄴ, ㄷ\\
(5) ᄀ, ㄴ, ㄷ