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  • Let <img alt="\mathrm{f(x)=[\tan x[\cot x]], x \in\left[\frac{\pi}{12}, \frac{\pi}{2}\right)}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bf%28x%29%3D%5B%5Ctan%20x%5B%5Ccot%20x

Let \mathrm{f(x)=[\tan x[\cot x]], x \in\left[\frac{\pi}{12}, \frac{\pi}{2}\right)} (where [.] represents the greatest integer function) then the number of points, where \mathrm{f(x)} is not continuous is

Option: 1

one
 


Option: 2

three
 


Option: 3

zero
 


Option: 4

infinite


Answers (1)

\mathrm{ \begin{aligned} & {[\cot x]=\cot x \text { if } \cot x \in I} \\ & \quad<\cot x \text { if } \cot x \notin I \\ & \Rightarrow[\tan x[\cot x]]=1 \text { if } \cot x \in I=0 \text { if } \cot x \notin I \end{aligned} }

so, points of discontinuity are those where \mathrm{\cot x \in I \, \, now \, \, \frac{\pi}{12} \leq x<\frac{\pi}{2}}

\mathrm{\Rightarrow 0<\cot x \leq 2+\sqrt{3}} Number of points of discontinuity is 3 .

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Kshitij

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