Question

Let $f\left(x\right)$ be a polynomial function of second degree. If $f\left(1\right)=f(-1)$ and $a,b,c\in A.P$ then $f{}^{\prime }\left(a\right),f{}^{\prime }\left(b\right),f{}^{\prime }\left(c\right)$ are in

• A. $G.P$
• B. $H.P$
• C. $A.G.P$
• D. $A.P$

Answer 1

The correct option is D $A.P$

Let the polynomial be $f\left(x\right)=a{x}^2+bx+c$
Given $f\left(1\right)=f(-1)$

$\Rightarrow a+b+c=a-b+c$

$\Rightarrow b=0$
Then, $f\left(x\right)=a{x}^2+c$
Now, $f{}^{\prime }\left(x\right)=2ax$
$\therefore f{}^{\prime }\left(a\right)=2a^2,f{}^{\prime }\left(b\right)=2ab,f{}^{\prime }\left(c\right)=2ac$
As $a,b,c\in A.P$
$\Rightarrow a^2,ab,ac\in A.P$
$\Rightarrow 2a^2,2ab,2ac\in A.P$
$\Rightarrow f{}^{\prime }\left(a\right),f{}^{\prime }\left(b\right),f{}^{\prime }\left(c\right)\in A.P$

Hence, $f{}^{\prime }\left(a\right),f{}^{\prime }\left(b\right),f{}^{\prime }\left(c\right)$ are in A.P.