We have a function: f ( x , y , z ) = x cos &#x2061; ( &#x03C9; t + y + &#x03D5; 1 ) + z cos &#x2061; ( &#x03C9; t + y + &#x03D5; 2 ) .I want to solve the following maximization problems: max x , y , z max t f ( x , y , z ) or max x , y , z &#x222B; &lt; T &gt; f ( x , y , z ) d t .Surely, x and z are nonnegative, and y is in [ 0 , 2 &#x03C0; ).Can someone give me hints for solving the problem, or let me know some references to solve this types of problem?

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We have a function:  f  (  x  ,  y  ,  z  )  =  x  cos  &amp;#x2061;      (    &amp;#x03C9;        t    +    y    +          &amp;#x03D5;      1        )    +  z  cos  &amp;#x2061;      (    &amp;#x03C9;        t    +    y    +          &amp;#x03D5;      2        )    .I want to solve the following maximization problems:      max          x      ,      y      ,      z                  max      t        f    (    x    ,    y    ,    z    )      or        max          x      ,      y      ,      z            &amp;#x222B;          &amp;lt;      T      &amp;gt;        f  (  x  ,  y  ,  z  )  d  t  .Surely,   x and   z are nonnegative, and   y is in   [  0  ,  2  &amp;#x03C0;  ).Can someone give me hints for solving the problem, or let me know some references to solve this types of problem?

candydulce168nlid · Accepted Answer

The first problem, the pointwise maximisation      max          x      ,      y      ,      z            max          t        f  (  x  ,  y  ,  z  )has not a solution: indeed, whatever   t and   y you select, the target function will diverge as   x and   →  ∞, as the two cosine functions will return values bounded by 0 and 1, and their argument do not depend on   x  ,  z.A similar line of thought is valid for the second problem too.Whatever candidate parameters   x  ,  y  ,  z you consider, you could increase the target function by increasing say   x, which does not affect the period.For the problem to be interesting, some constraint on   x  ,  z might be considered.In any case, both the problem are akin to maximising a function over three or four unknowns (as you can always solve the integral, for the second case): setting partial derivative to zero would be a good starting point. You woul find stationary points, and then you would have to check the stationary points corresponds to maxima.

We have a function: <br. f ( x , y , z ) = x cos ⁡<!--

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