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]

"> 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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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

Joint Entrance Exam - Main (JEE Main) Mathematics in Joint Entrance Exam - Main (JEE Main) . 10 months ago

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[b] \[f'(x)>0\] 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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