If x + y + z + w = 5 then the minimum value of x 2 cot ⁡ ( 9 ° ) + y 2 cot ⁡ ( 27 ° ) + z 2 cot ⁡ ( 63 ° ) + w 2 cot ⁡ ( 81 ° ) is?I used the A . M . ≥ G . M . rule. ( ( x y z w ) 2 . cot ⁡ ( 9 o ) tan ⁡ ( 9 o ) cot ⁡ ( 27 o ) tan ⁡ ( 27 o ) ) 1 / 4 ≤ x 2 cot ⁡ ( 9 ° ) + y 2 cot ⁡ ( 27 ° ) + z 2 cot ⁡ ( 63 ° ) + w 2 cot ⁡ ( 81 ° ) 4 This reduces to : 4 ( x y z w ) 1 / 2 ≤ x 2 cot ⁡ ( 9 ° ) + y 2 cot ⁡ ( 27 ° ) + z 2 cot ⁡ ( 63 ° ) + w 2 cot ⁡ ( 81 ° ) − − − − ( 1 )Now, I applied the same thing to the first equation: x + y + z + w 4 ≥ ( x y w z ) 1 / 4 5 4 ≥ ( x y w z ) 1 / 4 25 16 ≥ ( x y w z ) 1 / 2 − − − − ( 2 )Now I do not know what to do. From equation (2), I get the minimum value of ( x y w z ) 1 / 2 as − ∞ (Am I right over here ?). In a nutshell, I am confused what to do next.

Question

If   x  +  y  +  z  +  w  =  5 then the minimum value of       x    2    cot  ⁡  (  9      °    )  +      y    2    cot  ⁡  (  27      °    )  +      z    2    cot  ⁡  (  63      °    )  +      w    2    cot  ⁡  (  81      °    ) is?I used the   A  .  M  .  ≥  G  .  M  . rule.  (  (  x  y  z  w      )    2    .  cot  ⁡  (      9    o    )  tan  ⁡  (      9    o    )  cot  ⁡  (      27    o    )  tan  ⁡  (      27    o    )      )          1              /            4        ≤                       x        2            cot      ⁡      (      9              °            )      +              y        2            cot      ⁡      (      27              °            )      +              z        2            cot      ⁡      (      63              °            )      +              w        2            cot      ⁡      (      81              °            )        4  This reduces to :  4  (  x  y  z  w      )          1              /            2        ≤      x    2    cot  ⁡  (  9      °    )  +      y    2    cot  ⁡  (  27      °    )  +      z    2    cot  ⁡  (  63      °    )  +      w    2    cot  ⁡  (  81      °    )  −  −  −  −  (  1  )Now, I applied the same thing to the first equation:            x      +      y      +      z      +      w        4    ≥  (  x  y  w  z      )          1              /            4            5    4    ≥  (  x  y  w  z      )          1              /            4            25    16    ≥  (  x  y  w  z      )          1              /            2        −  −  −  −  (  2  )Now I do not know what to do. From equation (2), I get the minimum value of   (  x  y  w  z      )          1              /            2       as   −  ∞ (Am I right over here ?). In a nutshell, I am confused what to do next.

luthersavage6lm · Accepted Answer

By C-S      (          x      2        cot    ⁡    9          °        +          y      2        cot    ⁡    27          °        +          z      2        cot    ⁡    63          °        +          w      2        cot    ⁡    81          °        )        (    tan    ⁡          9              ∘              +    tan    ⁡          27              ∘              +    tan    ⁡          63              ∘              +    tan    ⁡          81              ∘              )    ≥  (  x  +  y  +  z  +  w      )    2    =  25The equality occurs for      (    x          cot      ⁡      9              °              ,    y          cot      ⁡      27              °              ,    z          cot      ⁡      63              °              ,    w          cot      ⁡      81              °              )        |        |        (          tan      ⁡              9                  ∘                      ,          tan      ⁡              27                  ∘                      ,          tan      ⁡              63                  ∘                      ,          tan      ⁡              81                  ∘                      )    ,Done!

I used the A.M.≥G.M. rule. ((xyzw)^2 . cot(9^o)tan(9^o)cot(27^o)tan(27^o))^{1/4}<=(x^2 cot (9°) + y^2 cot (27°) + z^2 cot (63°) + w^2 \cot (81°))/(4)

Answered question

Answer & Explanation

New Questions in Pre-Algebra