A) 2
B) 4
C) 6
D) 0
Correct Answer: B
Solution :
\[f(x)=\left\{ \begin{align} & 3-\left| \cos x-\frac{1}{\sqrt{2}} \right|,\,\,\,\,\,|\sin x|<\frac{1}{\sqrt{2}} \\ & 2+\left| \cos x+\frac{1}{\sqrt{2}} \right|,\,\,\,\,\,\left| \sin x \right|\,\ge \frac{1}{\sqrt{2}} \\ \end{align} \right.\] \[=\left\{ \begin{matrix} 3-\left| \cos x-\frac{1}{\sqrt{2}} \right|, & |\cos x|\,>\frac{1}{\sqrt{2}} \\ 2+\left| \cos x+\frac{1}{\sqrt{2}} \right|, & |\cos x|\,\le \frac{1}{\sqrt{2}} \\ \end{matrix} \right.\] \[=\left\{ \begin{matrix} 3-\cos x+\frac{1}{\sqrt{2}}, & |\cos x|\,>\frac{1}{\sqrt{2}} \\ 2+\cos x+\frac{1}{\sqrt{2}}, & |\cos x|\,\le \frac{1}{\sqrt{2}} \\ \end{matrix} \right.\] \[\therefore \]\[f(x)\]discontinuous at \[|\cos x|\,=\,\frac{1}{\sqrt{2}}\] or \[x=\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4}\]
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