Assignment -1 Question - None

Question

The value of x satisfies the inequality [tan^-1x]² - 2 [tan^-1x] - 3 $\leq$ 0, where [$\cdot$] represent greatest integer function, then x lies between

Accepted Answer

We have
[tan^-1x]² - 2 [tan^-1x] - 3 $\leq$ 0
Put [tan^-1x] = $\alpha$
$\Rightarrow$ $\alpha^{2}$ - 2$\alpha$ - 3 $\leq$ 0
$\Rightarrow \alpha^{2}+\alpha$ - 3$\alpha$ - 3 $\leq$ 0
$\Rightarrow$ $\alpha$ ($\alpha$ + 1) - 3($\alpha$ + 1) $\leq$ 0
$\Rightarrow(\alpha+1)(\alpha-3) \leq 0$

$\Rightarrow$ -1 $\leq$ $\alpha$ $\leq$ 3
$\Rightarrow$ -1 $\leq$ [tan^-1x] $\leq$ 3
$\Rightarrow$ -1 $\leq$ tan^-1x < 4
$\Rightarrow$ tan(-1) $\leq$ x < tan 4 < $\infty$
$\Rightarrow$ -tan 1 $\leq$ x < $\infty$
$\Rightarrow$ x $\in$ [-tan 1, $\infty$)

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