Let f(x) be a polynomial function of second degree and f(1) = f(-1). If a, b, c are in AP, then f'(a), f'(b), f'(c) are in:

Concept:

Arithmetic Progression (AP): The series of numbers where the difference of any two consecutive terms is the same, is called an Arithmetic Progression.

• If three numbers a, b and c are in AP, then b - a = c - b ⇒ 2b = a + c.

Geometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.

• If three numbers a, b and c are in GP, then\(\rm \dfrac ba=\dfrac cb\) ⇒ b2 = ac.

Harmonic Progression (HP): The series of numbers where the reciprocals of the terms are in Arithmetic Progression, is called a Harmonic Progression.

• If three numbers a, b and c are in HP, then \(\rm \dfrac{1}{a}+\dfrac{1}{c}=\dfrac{2}{b}\).

Calculation:

Let's say that the second degree polynomial is f(x) = px² + qx + r.

From the given information:

f(1) = f(-1)

⇒ p(1)2 + q(1) + r = p(-1)2 + q(-1) + r

⇒ p + q = p - q

⇒ 2q = 0

⇒ q = 0

∴ f(x) = px2 + 0(x) + r

⇒ f(x) = px2 + r

And, f'(x) = 2px.

Now, f'(a) = 2pa, f'(b) = 2pb and f'(c) = 2pc.

Since a, b and c are in AP, we have:

2b = a + c.

Multiplying by 2p, we get:

(2p)2b = (2p)a + (2p)c

⇒ 2(2pb) = 2pa + 2pc

⇒ 2pa, 2pb and 2pc are in AP.

⇒ f'(a), f'(b) and f'(c) are in AP.