The correct options are
A f(x) is continuous at x = 0
B f(x) is differentiable at x = 0
C f′(x) is continuous on R
f(0+)=f(0−)=f(0)
f(0+)=f(0−)=f(0)=0
So, f(x) is continuous at x=0
f′(x)={3x2,x<02x,x≥0
f′(0+)=f′(0−)=0,f′(0)=0
So, f(x) is differentiable and continuous at x=0 and f′(x) is continuous but non differentiable at x=0. Hence, f′′(x) will not be defined at x=0
f′′(x)={6x,x<02,x>0
f"(0+)=2f"(0−)=0
So, f′′(x) is not defined at x=0