A) \[\operatorname{f}\left( g\left( x\,+1 \right) \right)>f\left( g\left( x+5 \right) \right)\]
B) \[\operatorname{f}(g\left( x \right))<f\left( g\left( f(x+2 \right) \right)\]
C) \[\operatorname{g}\left( f(x \right))<g\left( f\left( x+2 \right) \right)\]
D) \[\operatorname{g}\left( f(x \right))<g\left( f\left( x-2 \right) \right)\]
Correct Answer: A
Solution :
Given, \[\operatorname{f}'\,(x) < 0 \,and \,g' (x) > 0\] therefore g(x) is an increasing function and f(x) is a decreasing function \[\therefore \,\,x+1<x+5~\,\,\,\Rightarrow \,\,g(x)<g\left( x+1 \right)<g\left( x+5 \right)\] \[\Rightarrow \,\,\,f(g(x+1))>f(g(x+5))\] Again \[\operatorname{x}<x+1\, \,\,\,\,\Rightarrow \,\,g\left( x+1 \right)\] \[\Rightarrow \,\,\,f\left( g\left( x \right) \right)>f\left( g\left( x+1 \right) \right)\] \[\operatorname{x}<x+2\,\,\Rightarrow \,\,f(x)>f\left( x+2 \right)\] \[\Rightarrow \,\,\,g\left( f\left( x \right) \right)>g\left( f\left( x+1 \right) \right)\] \[\operatorname{x}>x-2\,\,\Rightarrow \,\,\,f\left( x \right)<f\left( x\,-2 \right)\] \[\Rightarrow \,\,\,g\left( f\left( x \right) \right)<g\left( f\left( x\,-2 \right) \right)\]
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