Find whether the function f\left ( x \right )= \cos \left ( 2x+\frac{\pi }{4} \right );  is increasing or decreasing in the interval \frac{3\pi }{8}< x< \frac{5\pi }{8}\cdot

 

 

 

 
 
 
 
 

Answers (1)

f\left ( x \right )= \cos \left ( 2x+\frac{\pi }{4} \right )
\Rightarrow {f}'\left ( x \right )= -2\sin \left ( 2x+\frac{\pi }{4} \right )
As \frac{3\pi }{8}< x< \frac{5\pi }{8}\; \; \Rightarrow \; \frac{3\pi }{4}< 2x< \frac{5\pi }{4}
\Rightarrow \pi < 2x+\frac{\pi }{4}< \frac{3\pi }{2}
so clearly {f}'\left ( x \right )= -2\sin \left ( 2x+\frac{\pi }{4} \right )> 0 \, as
\left ( 2x+\frac{\pi }{4} \right )\, \epsilon \,III          Quadrant
so f\left ( x \right ) increasing  in \frac{3\pi }{8}< x< \frac{5\pi }{8}
 

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