Question
If $${\log _{\frac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0,$$ then exhaustive range of values of $$x$$ is :
A.
$$\left( { - \infty ,\,2} \right) \cup \left( {3,\,\infty } \right)$$
B.
$$\left( {2,\,3} \right)$$
C.
$$\left( { - \infty ,\,1} \right) \cup \left( {1,\,2} \right) \cup \left( {2,\,\infty } \right)$$
D.
none of these
Answer :
$$\left( {2,\,3} \right)$$
Solution :
$$\eqalign{
& f\left( x \right) = {\log _{\frac{1}{2}}}\left( {{x^2} - 5x + 7} \right) > 0 \cr
& \Rightarrow {x^2} - 5x + 7 > 0,\,{x^2} - 5x + 7 < 1,\,x\, \in \,R \cr
& \Rightarrow {x^2} - 5x + 6 < 0 \cr
& \Rightarrow x\, \in \left( {2,\,3} \right) \cr} $$