Question

Let $f\left(x\right)=0$ be a cubic equation with positive and distinct roots $\alpha ,\beta ,\gamma$ such that $\beta$ is harmonic mean between the roots of $f^{\prime }\left(x\right)=0.$ If $r=\left[\frac{2\beta }{\alpha +\gamma }\right]+\left[\frac{2\alpha \gamma }{\alpha \beta +\beta \gamma }\right],$ then the value of $\sum _{i=1}^3{i}^r$ is
( Here, $[.]$ denotes the greatest integer function.)

Answer 1

Let $f\left(x\right)=x^3+a{x}^2+bx+c=0$
and $\alpha ,\beta ,\gamma$ be the roots of the above equation.
$\alpha +\beta +\gamma =-a$
$\alpha \beta +\beta \gamma +\gamma \alpha =b$
$\alpha \beta \gamma =-c$

$f^{\prime }\left(x\right)=3x^2+2ax+b$
Let $x_1,x_2$ be the roots of $f^{\prime }\left(x\right)=0$
$x_1+x_2=\frac{-2a}{3}$
$x_1{x}_2=\frac{b}{3}$
Given, $\beta$ is harmonic mean between $x_1$ and $x_2.$
$\Rightarrow \beta =\frac{2x_1{x}_2}{x_1+x_2}=\frac{2b}{-2a}=\frac{b}{-a}$
$\Rightarrow \beta =\frac{(\alpha \beta +\beta \gamma +\gamma \alpha )}{(\alpha +\beta +\gamma )}$
$\Rightarrow \alpha \beta +{\beta }^2+\beta \gamma =\alpha \beta +\beta \gamma +\gamma \alpha$
$\Rightarrow {\beta }^2=\alpha \gamma$
$\Rightarrow \alpha ,\beta ,\gamma$ are in G.P.

$r=\left[\frac{2\beta }{\alpha +\gamma }\right]+\left[\frac{2\alpha \gamma }{\alpha \beta +\beta \gamma }\right]$
$\Rightarrow r=\left[\frac{2\sqrt{\alpha \gamma }}{\alpha +\gamma }\right]+\left[\frac{2\alpha \gamma }{\beta (\alpha +\gamma )}\right]$
$\Rightarrow r=\left[\frac{G.M.}{A.M.}\right]+\left[\frac{H.M.}{G.M.}\right]=0+0=0$
$\{\because A.M.>G.M.>H.M.\}$
$\therefore \sum _{i=1}^r(i{)}^3=\sum _{i=1}^{3}1=1+1+1=3$