Find the range of the function f ( x ) = cos ⁡ ( x ) [ sin ⁡ ( x ) + sin 2 ⁡ ( x ) + 1 2 ]

Question

Find the range of the function   f  (  x  )  =  cos  ⁡  (  x  )  [  sin  ⁡  (  x  )  +            sin      2        ⁡    (    x    )    +          1      2        ]

boursecasa2je · Accepted Answer

f  (  x  )  −  cos  ⁡  x  sin  ⁡  x  =  cos  ⁡  x            1      2        +          sin      2        ⁡    x    ⇒  f  (  x      )    2    −  2  f  (  x  )  cos  ⁡  x  sin  ⁡  x  =      1    2        cos    2    ⁡  x  ⇒    4  f  (  x  )  cos  ⁡  x  sin  ⁡  x  =  2  f  (  x      )    2    −      cos    2    ⁡  x  ⇒  16  f  (  x      )    2        cos    2    ⁡  x  (  1  −      cos    2    ⁡  x  )  =  (  2  f  (  x      )    2    −      cos    2    ⁡  x      )    2    f  (  x      )    2    =  y  ,         cos    2    ⁡  x  =  t  :  16  y  t  (  1  −  t  )  =  (  2  y  −  t      )    2    ⇒  4      y    2    +  (  16      t    2    −  20  t  )  y  +      t    2    =  0  ⇒    8  y      y    ′    +  (  16      t    2    −  20  t  )      y    ′    +  (  32  t  −  20  )  y  +  2  t  =  0Here       y    ′    =            d      y              d      t      . In critical points             d      f      (      x      )              d      x        =  0  ⇒            d      y              d      t        =  0  (  32  t  −  20  )  y  +  2  t  =  0  ⇒  y  =  −      t          16      t      −      10        ⇒            4              t        2                    (      16      t      −      10              )        2              −            t      (      16              t        2            −      20      t      )              16      t      −      10        +      t    2    =  0  ⇒    4      t    2    −  t  (  16      t    2    −  20  t  )  (  16  t  −  10  )  +      t    2    (  16  t  −  10      )    2    =  0  ⇒  32      t    2    (  5  t  −  3  )  =  0  t  =  0 gives minimum   y  =  0.   t  =      3    5   gives maximum   y  =      3    2  Minimum   f  (  x  )  =  −            3      2       is obtained at   cos  ⁡  x  =  −            3      5      ,   sin  ⁡  x  =            2      5      Maximum   f  (  x  )  =            3      2       is obtained at   cos  ⁡  x  =            3      5      ,   sin  ⁡  x  =            2      5

Find the range of the function f(x)=cos(x)[sin(x)+sqrt(sin^2(x)+1/2)]

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