For the function f(x)=0 for x≤0, f(x)=x2logx for x>0
f’(0) exist at x=0
f’(0) does not exist
Lf’(0)=–1
Rf’(0)=1
Explanation for the correct option:
Finding limit on the given function.
Left hand limit,
Lf’(0)=f’(0-)=limh→0[f(0–h)–f(0)]h=limh→0[0–0]h=0
Right hand limit,
Rf’(0)=f’(0+)=limh→0[f(0+h)–f(0)]h=limh→0[h2logh]h=limh→0hlogh=0f’(0)=0
Since left hand and right hand limit exist at x=0
Therefore, f’(0) exist at x=0
Hence, correct option is option (A).