Question

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) and f(c) are in 

(A) G.P. 

(B) H.P. 

(C) AG.P. 

(D) AP. 

Answer 1

Answer is (D) AP

Let f(x) = px²+ qx + r

f(1) = f(-1) gives p + q + r = p - q + r

hence q = 0

Hence f(x) = px²+ r 

f'(x) = 2px ....(1)

Given a, b, c are in A.P.

hence 2pa, 2pb, 2pc will also be in AP.

or f"(a), f'(b), f'(c) will also be in AP.