Let ƒ(x) be polynomial function of second degree. If ƒ(1) =…

Maths Sequence and Series ( Progressions ) Arithmetic Progression Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 13, 2026

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

ƒ(1) = ƒ(–1) and a, b, c are in AP, then ƒ ′

, ƒ ′ AP

and ƒ ′ GP

are in - HP

Arithmetico-geometric progression

Text Solution

Verified by Experts

The correct answer is:

Let ƒ(x) = ax 2 + 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) = ax 2 + c

ƒ ′ (x) = 2 ax

∴ ƒ ′ = 2a 2 , ƒ ′ = 2ab, ƒ ′ = 2ac

Now, we take 2ƒ ′ = ƒ ′ + ƒ ′

⇒ 2.2ab = 2a 2 + 2ac

⇒ 2b = a + c

⇒ a, b, c are in AP

⇒ ƒ ′ , ƒ ′ , ƒ ′ are in AP.

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