Let f(x)=0 be a cubic equation with positive and distinct roots α,β,γ such that β is harmonic mean between the roots of f′(x)=0. If r=[2βα+γ]+[2αγαβ+βγ], then the value of 3∑i=1ir is ( Here, [.] denotes the greatest integer function.)

Let f(x)=0 be a cubic equation with positive and distinct roots α,β,

1.	Let f(x)=0 be a cubic equation with positive and distinct roots α,β,γ such that β is harmonic mean between the roots of f′(x)=0. If r=[2βα+γ]+[2αγαβ+βγ], then the value of 3∑i=1ir is ( Here, [.] denotes the greatest integer function.)
Answer» Let $f (x) = 0$ be a cubic equation with positive and distinct roots $α, β, γ$ such that $β$ is harmonic mean between the roots of $f^{'} (x) = 0.$ If $r = [\frac{2 β}{α + γ}] + [\frac{2 α γ}{α β + β γ}],$ then the value of $3 \sum i = 1 i^{r}$ is ( Here, $[.]$ denotes the greatest integer function.)