Letg:R→Rbe a differentiable function withg(0)=0,g′(0)=0andg′(1)≠0. Letf(x)=x|x|g(x),x≠00,x=0andh(x)=e|x|for allx∈R. Let(f⋅h)(x)denotesf(h(x))and(h⋅f)(x)denoteh(f(x)). Then which of the following is (are) true?

The correct options areAfis differentiable atx=0Dh⋅fis differentiable atx=0given g(0)=0g′(0)=0g′(1)≠1f(x)=−g(x)x<0g(x)x>00x=0For option (A) R.H.D=L.H.D(by applying basic differntiation definition).For option (B)h(x)=−e−x<0,exforx>0,0forx=0L.H.D is not equal to R.H.D in this case.For option (C)foh=f(h(u))=g(h(u)),h(u)>0foh 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−0since L.H.D =R.H.D it is differentiable