A) \[f\{g(x)\}\ge f\{g(0)\}\]
B) \[g\{f(x)\}\le g\{f(0)\}\]
C) \[f\{g(2)\}=7\]
D) None of these
Correct Answer: B
Solution :
\[f'(x)>0\] if \[x\ge 0\] and \[g'(x)<0\] if \[x\ge 0\] |
Let \[h(x)=f(g(x))\] |
then \[h'(x)=f'(g(x)).g'(x)<0\] if \[x\ge 0\] |
\[\therefore \,\,h(x)\] is decreasing function |
\[\therefore \,\,h(x)\le h(0)\] if \[x\ge 0\] |
\[\therefore \,\,f(g(x))\le f(g(0))=0\] |
But codomain of each function is \[[0,\infty )\] |
\[\therefore \,\,f(g(x))=0\] for all \[x\ge 0\] |
\[\therefore \,\,f(g(x))=0\] |
Also \[g(f(x))\le g(f(0))\] [as above] |
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