Question

Let ƒ(x) be a polynomial function of second degree. If f(1) = f(–1) and a, b, c are in A.P., then ƒ′(1), ƒ′(2), ƒ′(3) are in –

• A. G.P.
• B. H.P.
• C. Arithmetic Geometric progression
• D. A.P.

Answer 1

Answer: (D)

A.P.

Answer 2

Let ƒ(x) = ax² + bx + c

ƒ(1) = ƒ(–1) ⇒ a + b + c = a – b + c

⇒ 2b = 0

⇒ b = 0

∴ ƒ (x) = ax² + c ∴ ƒ′(x) = 2ax

∴ ƒ′(1) = 2a², ƒ′(2) = 2ab, ƒ′(3) = 2ac

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

∴ ƒ′(1),ƒ′(2), ƒ′(3) are also in A.P.