Number of real solution of Trigonometric EquationNumber of solution of the equation ( sin ⁡ x + cos ⁡ x + 2 ) 4 = 128 sin ⁡ ( 2 x ) ∀ x ∈ [ 0 , π 2 ] What i try sin ⁡ x + cos ⁡ x + 2 = 2 cos ⁡ ( x − π 4 ) + 2And put x − π 4 = t and t ∈ [ − π 4 , π 4 ] ( cos ⁡ t + 2 ) 4 = − 64 cos ⁡ ( 2 t )How do i solve it Help me

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Number of real solution of Trigonometric EquationNumber of solution of the equation   (  sin  ⁡  x  +  cos  ⁡  x  +  2      )          4        =  128  sin  ⁡  (  2  x  )    ∀  x  ∈            [        0  ,      π    2              ]      What i try  sin  ⁡  x  +  cos  ⁡  x  +  2  =      2    cos  ⁡            (        x  −      π    4              )        +  2And put       x    −          π      4        =    t   and       t    ∈                  [              −          π      4        ,          π      4                      ]                        (        cos  ⁡  t  +      2                      )              4    =  −  64  cos  ⁡  (  2  t  )How do i solve it Help me

Taniyah Lin · Accepted Answer

From arithmetic Geometric Inequality            sin      ⁡      x      +      cos      ⁡      x      +      2        4    ≥            [        sin  ⁡  x  ⋅  cos  ⁡  x                    ]                            1        4              (  sin  ⁡  x  +  cos  ⁡  x  +  2      )    4    ≥  128  ⋅  sin  ⁡  2  xEquality hold when   sin  ⁡  x  =  cos  ⁡  x  =  1  =  1

Jadon Johnson · Accepted Answer

Let   sin  ⁡      x    +  cos  ⁡      x    =  tThus, by C-S      |    t      |    ≤      (    1    +    1    )    (          sin      2        ⁡    x    +          cos      2        ⁡    x    )    =      2  and we need to solve that   f  (  t  )  =  0  , where  f  (  t  )  =  (  t  +  2      )    4    −  128  (      t    2    −  1  )  .But      f    ″    (  t  )  =  12  (  t  +  2      )    2    −  256  ≤  12  (      2    +  2      )    2    −  256  &amp;lt;  0  ,which says that   f is a concave function on   [  0  ,      2    ] and since   f  (  0  )  &amp;gt;  0 and   f  (      2    )  &amp;gt;  0,we see that our equation has no real roots.

Number of real solution of Trigonometric Equation Number of solution of the equation (sin x+ cos x+2)^(4)=128 sin(2x) AA x in [0,(pi)/(2)]

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