JEE Main & Advanced Sample Paper JEE Main - Mock Test - 14

Let f and g be functions from the interval \[[0,\infty )\]to the interval \[[0,\infty )\] f being an increasing and g being a decreasing function. If \[f\{g(0)\}=0\] then

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]