Let f(x) be a polynomial function of second degree

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

  1. G.P.
  2. A.P.
  3. A.P. - G.P.
  4. H.P.

Answer

Let the polynomial be f(x) = ax2 + bx + c

Given, f(1) = f(–1)

This implies, b = 0

Therefore, f(x) = ax2 + c

f'(x) = 2ax

Therefore, f'(a) = 2a2, f'(b) = 2ab, f'(c) = 2ac

As a, b, c are in A.P, a2, ab, ac are in A.P.

2a2, 2ab, 2ac are in A.P.

The correct option is B.