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Question

Let g:RR be a differentiable function with g(0)=0,g(0)=0 and g(1)0. Let f(x)={x|x|g(x),x00,x=0 and h(x)=e|x| for all xR. Let (fh)(x) denotes f(h(x)) and (hf)(x) denote h(f(x)). Then which of the following is (are) true?

A
f is differentiable at x=0
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B
h is differentiable at x=0
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C
fh is differentiable at x=0
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D
hf is differentiable at x=0
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Solution

The correct options are
A f is differentiable at x=0
D hf is differentiable at x=0
given g(0)=0 g(0)=0g(1)1
f(x)=g(x)x<0g(x)x>00x=0
For option (A) R.H.D=L.H.D(by applying basic differntiation definition).
For option (B) h(x)=ex<0,exforx>0,0forx=0
L.H.D is not equal to R.H.D in this case.
For option (C)
foh=f(h(u))=g(h(u)), h(u)>0
foh is not differentiable at x=0 (since L.H.D is not equal to R.H.D)
For option (D)R.H.Dlimt0h(f(t))h(f(0))t=limt0h(g(t))h(0)t=0L.H.Dlimt0h(f(t))h(f(0))t=limt0h(g(t))1t0
since L.H.D =R.H.D it is differentiable

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