First slide

Arithmetic progression

Question

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

Moderate

A

G.P.
B

H.P.
C

A.G.P.
D

A.P.

Solution

Let $f (x) = A x^{2} + B x + C .$
$f (1) = f (- 1) \Rightarrow A + B + C \Rightarrow A - B + C$
$\begin{array}{lrl} \Rightarrow 2 B = 0 \Rightarrow B = 0 . \\ ∴ f (x) = A x^{2} + C \Rightarrow f^{'} (x) = 2 A x \end{array}$
As $a, b, c$ are in A.P., $2 A a, 2 A b, 2 A c$ are in A.P., i.e., $f' (a), f' (b), f' (c)$ are in A.P.

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