Question

$\alpha ,\beta ,\gamma$ and $\delta$ are the zeros of a polynomial function $f\left(x\right)$ of degree $4.$ If $\alpha ,\beta ,\gamma$ and $\delta$ are in A.P., then $f^{\prime \prime \prime }\left(\alpha \right),f^{\prime \prime \prime }\left(\beta \right),f^{\prime \prime \prime }\left(\gamma \right)$ and $f^{\prime \prime \prime }\left(\delta \right)$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A A.P.

Let $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$
$f^{\prime }\left(x\right)=4a{x}^3+3b{x}^2+2cx+d$
$f^{\prime \prime }\left(x\right)=12a{x}^2+6bx+2c$
$f^{\prime \prime \prime }\left(x\right)=24ax+6b$

$\because \alpha ,\beta ,\gamma ,\delta$ are in A.P.,
$\therefore 24a\alpha ,24a\beta ,24a\gamma ,24a\delta$ are also in A.P.
$\Rightarrow 24a\alpha +6b,24a\beta +6b,24a\gamma +6b,24a\delta +6b$ are in A.P.
$\therefore f^{\prime \prime \prime }\left(\alpha \right),f^{\prime \prime \prime }\left(\beta \right),f^{\prime \prime \prime }\left(\gamma \right),f^{\prime \prime \prime }\left(\delta \right)$ are in A.P.