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 $f^{\prime}(a), f^{\prime}(b)$ and $f^{\prime}(c)$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. arithmetico-geometric progression

Answer 1

Answer: (A)

Let $f(x)=A x^2+B x+C$
$ \therefore f(1) =A+B+C $
and $f(-1) =A-B+C$
$ \because f(1) =f(-1) $
$ \Rightarrow A+B+C =A-B+C $
$\Rightarrow B =0$
$ \therefore f(x) =A x^2+C$
$ \Rightarrow f^{\prime}(x) =2 A x $
$ \Rightarrow f^{\prime}(a) =2 A a, f^{\prime}(b)=2 A b$
and $ f^{\prime}(c) =2 A c$
Also, $a, b, c$ are in A.P. .
$\therefore 2 A a, 2 A b, 2 A c$ are in A.P.
$\Rightarrow f^{\prime}(a), f^{\prime}(b), f^{\prime}(c)$ are in A.P.