A) \[\left[ -\frac{\pi }{3},\,\,-\frac{\pi }{6} \right]\]
B) \[\left[ \frac{\pi }{6},\,\,\frac{\pi }{3} \right]\]
C) \[\left[ \frac{1}{2}-\frac{\pi }{3}\left( 1+\frac{\pi }{3} \right),\,\,\frac{\sqrt{3}}{2}-\frac{\pi }{6}\left( 1+\frac{\pi }{6} \right) \right]\]
D) \[\left[ \frac{1}{2}+\frac{\pi }{3}\left( 1-\frac{\pi }{3} \right),\,\,\frac{\sqrt{3}}{2}+\frac{\pi }{6}\left( 1-\frac{\pi }{6} \right) \right]\]
Correct Answer: C
Solution :
Since,\[f(x)\]is a continuous decreasing function on\[\left[ \frac{\pi }{6},\,\,\frac{\pi }{3} \right]\]. \[\therefore \]\[f(x)\]Attains every value between\[\left[ \frac{\pi }{6},\,\,\frac{\pi }{3} \right]\]its minimum value \[ie\], \[f\left( \frac{1}{3} \right)=\frac{1}{2}-\frac{\pi }{3}\left( 1+\frac{\pi }{3} \right)\] and maximum value is \[f\left( \frac{\pi }{6} \right)=\frac{\sqrt{3}}{2}-\frac{\pi }{6}\left( 1+\frac{\pi }{6} \right)\]
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