Question

Let ƒ(x) be polynomial function of second degree. If

ƒ(1) = ƒ(–1) and a, b, c are in AP, then ƒ¢(1), ƒ¢(2) and ƒ¢(3) are in –

• A. AP
• B. GP
• C. HP
• D. Arithmetic-geometric progression

Answer 1

Answer: (A)

AP

Answer 2

Let ƒ(x) = ax² + bx + c

ƒ(1) = a + b + c

ƒ(–1) = a – b + c

Since, ƒ(1) = ƒ(–1)

̃ a + b + c = a – b + c

̃ 2b = 0 ̃ b = 0

So ƒ(x) = ax² + c

Ģ(x) = 2 ax

\ ƒ¢(1) = 2a², ƒ¢(2) = 2ab, ƒ¢(3) = 2ac

Now, we take 2Ģ (2) = Ģ(1) + Ģ(3)

̃ 2.2ab = 2a² + 2ac

̃ 2b = a + c

̃ a, b, c are in AP

̃ ƒ¢(1), ƒ¢(2), ƒ¢(3) are in AP.