Let f(x)=x|x| and g(x)= x . Statement-1 : gof is differentiable at x=0 and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at x=0 .

Step-by-step solution

aligned & f(x)=x|x| and g(x)= x & g f(x)= (x|x|)= array cc - x^2 & , x < 0 x^2 & , x 0 array . & (gof )^ (x)= array cc -2 x x^2 & , x < 0 2 x x^2 & , x 0 array . aligned Clearly, L(gof )' (0)=0= R ( gof )^ (0) gof is differentiable at x=0 and also its derivative is continuous at x=0 aligned & Now, (gof)" ( x )= array cc -2 x ^2+4 x ^2 x ^2 & , x < 0 2 x ^2-4 x ^2 x ^2 & , x 0 array . & L ( gof )^ (0)=-2 and R ( gof )^ (0)=2 & L ( gof )^ (0) R ( gof )^ (0) aligned g o f(x) is not twice differentiable at x=0 .