<s>[INST] Greatest integer function problem If $\alpha$ is real root of equation $$x^5 -x^3 +x-2$$ , then [$\alpha$^6 ] is equal to ? Here [.] represents greatest integer function . Any hint please!
 [/INST] $x^5 - x^3 + x = \frac {x(x^6 + 1)}{x^2 + 1}\\
x(x^4 - x^2  +1) = \frac {x}{x^2+1}(x^6 + 1)\\
\frac {x}{x^2+1}(x^6 + 1) = 2$
We can conclude that $x> 0$
as $(x^6 + 1)> 0$ we need $\frac {x}{x^2+1} > 0$
$\frac {x}{x^2+1} < \frac 12$ for all x
$(x^6 + 1) > 4\\
\alpha^6 > 3$
But is it possible for $\alpha^6 > 4$?
$f(x) = x^5 - x^3 + x - 2$ is an increasing function, 
If $f(\alpha) > 0$ then it is not possible for there to be any root to the right of $x^6 = 4$
$\lfloor \alpha^6\rfloor = 3$
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