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Question

Let f:RR,g:RR and h:RR be differentiable functions such that f(x)=x3+3x+2,g(f(x))=x and h(g(g(x)))=x for all x ε R. Then


A

g(2)=115

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B

h(1)=666

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C

h(0)=16

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D

h(g(3))=36

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Solution

The correct option is C

h(0)=16


f(x)=x3+3x+2f(x)=3x2+3Also f(0)=2,f(1)=6,f(2)=16,f(3)=38,f(6)=236And g(f(x))=xg(2)=0,g(6)=1,g(16)=2,g(38)=3,g(236)=6


(a)g(f(x))=xg(f(x)).f(x)=1For g(2),f(x)=2x=0
Putting x=0, we get g(f(0))f(0)=1
g(2)=13

(b) h(g(g(x)))=xh(g(g(x))).g(g(x)).g(x)=1For h(1) we need g (g(x)) = 1
g(x)=6x=236 Putting x = 236, we get h[g(g(236))]=1g(g(236)).g(236)h(g(6))=1g(6).g(236)h(1)=1g(f(1)).g(f(6))=f(1).f(6)=6×111=666


(c) h[g(g(x))]=xFor h(0),g(g(x))=0g(x)=2x=16
Putting x = 16, we get
h(g(g(16)))=16h(0)=16


(d)h[g(g(x))]=xFor h(g(3)),we need g(x)=3x=38
Putting x = 38, we get
h[g(g(38))]=38h(g(3))=38


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