Let f(x) be a polynomial function of second degree. If f(1)=f(-1) and a,b,c are in A.P., then f'(a), f'(b), f'(c) are in

Let f(x)=Ax2+Bx+C.f(1)=f(−1)⇒A+B+C⇒A−B+C⇒         2B=0⇒B=0.∴        f(x)=Ax2+C⇒f′(x)=2AxAs a,b,c are in A.P., 2Aa, 2Ab, 2Ac are in A.P., i.e., f'(a), f'(b), f'(c) are in A.P.