Let g:R→R be a differentiable function with g(0)=0,g′(0)=0 and g′(1)≠0. Let f(x)={x|x|g(x),x≠00,x=0 and h(x)=e|x| for all x∈R. Let (f⋅h)(x) denotes f(h(x)) and (h⋅f)(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
f⋅h is differentiable at x=0
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D
h⋅f is differentiable at x=0
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Solution
The correct options are Af is differentiable at x=0 Dh⋅f 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)=−e−x<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.Dlimt→0h(f(t))−h(f(0))t=limt→0h(g(t))−h(0)t=0L.H.Dlimt→0h(f(−t))−h(f(0))−t=limt→0h(−g(−t))−1−t−0 since L.H.D =R.H.D it is differentiable