Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$. (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is false. (3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
Consider a quadratic equation $a x ^ { 2 } + b x + c = 0$, where $2 a + 3 b + 6 c = 0$ and let $g ( x ) = a \frac { x ^ { 3 } } { 3 } + b \frac { x ^ { 2 } } { 2 } + c x$. Statement 1: The quadratic equation has at least one root in the interval $( 0,1 )$. Statement 2: The Rolle's theorem is applicable to function $g ( x )$ on the interval $[ 0,1 ]$.\\
(1) Statement 1 is false, Statement 2 is true.\\
(2) Statement 1 is true, Statement 2 is false.\\
(3) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.\\
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.