If the derivative of the function f(x)=ax2+b,    x<−1bx2+ax+4,    x≥−1 is everywhere continuous, then

f(x)=ax2+b,              x<−1bx2+ax+4,  x≥−1⇒f'x=2ax,            x<−12bx+a,    x≥−1Since f(x) is differentiable at x=−1,∴fx is continuous at x=−1∴limx→1−f(x)=limx→1+f(x)⇒a+b=b−a+4⇒a=2             ....(1)Also, since f'(x) is continuous, therefore, f'(x) is continuous at x=−1.∴limx→−1−f'(x)=limx→−1+f'(x)⇒−2a=−2b+a                          ....(2)Using (1) in (2), we get, b=3∴a=2, b=3.