Applications Of Derivatives Question 16
Question: 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
Options:
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
Show Answer
Answer:
Correct Answer: B
Solution:
[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 a 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]
BETA
