Question

The complete solution set of the inequality ${\left[{\cot }^{-1}x\right]}^2-6\left[{\cot }^{-1}x\right]+9\le 0$, where $[\cdot ]$ dentoes greatest integer function is

• A. $[-\infty ,\cot 3]$
• B. $[\cot 3,\cot 2]$
• C. $[\cot 3,\infty )$
• D. $noneofthese$

Answer 1

The correct option is A $[-\infty ,\cot 3]$

$\left[{\cot }^{-2}x{]}^{-}6\right[{\cot }^{-2}x]+9\le 0$

$\left(\right[{\cot }^{-2}x]-3{)}^2\le 0$

It's a square so it cannot be less than zero, but it can be equal to zero so

$\left(\right[{\cot }^{-1}x]-3{)}^2=0$

$\left[{\cot }^{-1}x\right]-3=0$

$\left[{\cot }^{-1}x\right]=3$

so

$3\le {\cot }^{-3}x<4$

graph of ${\cot }^{-1}x$

It can be seen that $3.14\le co{t}^{-1}x\le 3$

for $-\infty <x\le \cot 3$

so $x\in (-\theta ,\cot 3]$.