Question

If $a$ is an integer satisfying $|a|\le 4-|\left[x\right]|,$ where $x$ is a real number for which $2x{\tan }^{-1}x$ is greater than or equal to $\ln (1+x^2),$ then the number of maximum possible values of $a$ is (are)
(where $[.]$ represents the greatest integer function)

Answer 1

Let $y=2x{\tan }^{-1}x-\ln (1+x^2)$
$y^{\prime }=2{\tan }^{-1}x+\frac{2x}{1+x^2}-\frac{2x}{1+x^2}$
$y^{\prime }>0\forall x\in R^{+},y^{\prime }<0\forall x\in R^{-}$
or $y\ge 0\forall x\in R$
Therefore, $4-|\left[x\right]|$ takes the values $0,1,2,3,4(\therefore |a|\le 4-|[x]|$)

$|a|\le 4-|\left[x\right]|$) is satisfied by $a=0,\pm 1,\pm 2,\pm 3,\pm 4$.
Therefore, number of values of $a$ is $9$.