Find Nearest Integer to $\dfrac{1}{k^3-2009}$

[anemone]

Let $k$ be the largest root of $x^4+1-2009x=0$. Find the nearest integer to $\dfrac{1}{k^3-2009}$.

 

[kaliprasad]

Let $k$ be the largest root of $x^4+1-2009x=0$. Find the nearest integer to $\dfrac{1}{k^3-2009}$.

Spoiler

x(x^3-2009) = -1

so 1/(x^3-2009) = - x

so we need to find the nearest integer to -x

now largest x is between 12.6 and 12.7(

method to compute x^4 = 2009 x, ignoring 1 and so x^3 = 2009 and then check )

so ans is - 13

 

[anemone]

Well done, kaliprasad!