Let g   : R → R be a differentiable function with g 0 = 0 ,         g ′…

Let g :R→R be a differentiable function with g0=0,    g′0=0 and g′1 ≠0.  Let fx= x|x| gx,x ≠00,x=0 and hx=e|x| for all x∈R. Let foh (x) denote fhxand (hof) (x) denote hfx .Then which of the following is (are) true?
  1. f is differentiable at x=0
  2. h is differentiable at x=0
  3. f ₀ h is differentiable at x=0
  4. h ₀ f is differentiable at x=0

Solution

fx=gxx>00x=0-gxx<0
LHD: limx→0-⁡-g′x=-limx→0-⁡g′x=-0=0
RHD: limx→0+⁡g′x=g′0+=0
Hence fx is differentiable at x=0
hx=ex=exx≥0e-xx<0
LHD: limx→0-⁡-e-x=-1
RHD: limx→0+⁡ex=1
Hence hx is not differentiable at x=0
fhx=exexgex
fhx=gexx≥0ge-xx<0
LHD: limx→0+⁡g′e-x×-e-x =g′1×-1=-g′1
RHD: limx→0+⁡g′ex×ex=g′1
Hence foh is non-differentiable at x=0
hfx=1x=0exxgxx≠0
=1x=0egxx≠0
LHD: limx→0-⁡egx×±g′x=eg0-×±g′0-=0
RHD: limx→0+⁡egx×±g′x=eg0+×±g′0+=0

Asked in: JEE Advanced 2015 (Paper 1)

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