Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then (1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$ (2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$ (3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$ (4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$
Let $g : R \rightarrow R$ be a non constant twice differentiable such that $g ^ { \prime } \left( \frac { 1 } { 2 } \right) = g ^ { \prime } \left( \frac { 3 } { 2 } \right)$. If a real valued function $f$ is defined as $f ( x ) = \frac { 1 } { 2 } [ g ( x ) + g ( 2 - x ) ]$, then\\
(1) $f ^ { \prime \prime } ( x ) = 0$ for atleast two $x$ in $( 0,2 )$\\
(2) $f ^ { \prime \prime } ( x ) = 0$ for exactly one $x$ in $( 0,1 )$\\
(3) $f ^ { \prime \prime } ( x ) = 0$ for no $x$ in $( 0,1 )$\\
(4) $f ^ { \prime } \left( \frac { 3 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) = 1$