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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 xR. 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 options are
B h(1)=666
C h(0)=16
f(x)=x3+3x+2
f(x)=3x2+3
Now, we have g(f(x))=xg(f(x)).f(x)=1
We need to find g(2)f(x)=2x=0
g(2).f(0)=1g(2)=13

Now, we have h(g(g(x)))=x
Replacing x by f(x) in the above expression:
h(g(g(f(x))))=f(x)h(g(x))=f(x) ...[1]
Replacing x by f(x) in the above expression:
h(g(f(x)))=f(f(x))h(x)=f(f(x))h(x)=f(f(x))f(x)

h(0)=f(f(0))=f(2)=16
h(1)=f(f(1))f(1)=f(6)×6h(1)=111×6=666

h(g(3))=f(3) (From [1])h(g(3))=33+3×3+2=38

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