The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is (a) $( - \infty , 1 )$ (b) $( - \infty , - 1 )$ (c) $( 1 , \infty )$ (d) $( 3 , \infty )$
(B) Now, $x ^ { 2 } - 2 x - 3 > 0$ $\Rightarrow ( x - 1 ) ^ { 2 } > 4$ $\Rightarrow x > 3$ or $x < - 1$ $f ( x ) = \left\{ \log \left( x ^ { 2 } - 2 x - 3 \right) \right\} / \log ( 1 / 2 ) = - \left\{ \log \left( x ^ { 2 } - 2 x - 3 \right) \right\} / \log ( 2 )$ $f ^ { \prime } ( x ) = - ( 1 / \log 2 ) ( 2 x - 2 ) / \left( x ^ { 2 } - 2 x - 3 \right)$ $f ^ { \prime } ( x ) = - ( 1 / \log 2 ) 2 ( x - 1 ) / \{ ( x + 1 ) ( x - 3 ) \}$ To be monotonically increasing $f ^ { \prime } ( x )$ must be $> 0$. Options (c) and (d) cannot be true. Option (a) cannot be true.
The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is\\
(a) $( - \infty , 1 )$\\
(b) $( - \infty , - 1 )$\\
(c) $( 1 , \infty )$\\
(d) $( 3 , \infty )$