Question

Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a₁, a₂, a₃ are in AP, then f’(a₁), f’(a₂), f’(a₃) are in ______.

विकल्प

• AP

• GP

• HP

• None of these

Answer 1

Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a₁, a₂, a₃ are in AP, then f’(a₁), f’(a₂), f’(a₃) are in AP.

Explanation:

Let f(x) = ax² + bx + c

f(1) = f(–1)

`\implies` a + b + c = a – b + c

`\implies` b = 0

∴ f(x) = ax² + c

Differentiating w.r.t.x, f’(x) = 2ax

∴ f’(a₁) = 2aa₁,

f’(a₂) = 2aa₂

and f’(a₃) = 2aa₃

Assume that, f’(a₁), f’(a₂) and f’(a₃) are in AP, then 

2f’(a₂) = f’(a₁) + f’(a₃)

`\implies` 2 . 2aa₂ = 2aa₁ + 2aa₃

`\implies` 2a₂ = a₁ + a₃

So, a₁, a₂, a₃ are also in AP.

∴ f’(a₁), f’(a₂), f’(a₃) are in AP.