I was trying to solve the problem A maximization problem when I ask myself if the general problem m a x i m i z e f ( X ) p + g ( X ) p s . t . X ∈ K ⊆ R m × n , is equivalent to m a x i m i z e f ( X ) + g ( X ) s . t . X ∈ K ⊆ R m × n , when the scalar functions f ( X ) and g ( X ) are nonnegative on K, and p &gt; 0.Is this true? If not, how to find a counterexample?

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I was trying to solve the problem A maximization problem when I ask myself if the general problem                    m        a        x        i        m        i        z        e                f        (                  X                          )          p                +        g        (                  X                          )          p                                    s        .        t        .                          X                ∈        K        ⊆                              R                                m            ×            n                          ,            is equivalent to                    m        a        x        i        m        i        z        e                f        (                  X                )        +        g        (                  X                )                            s        .        t        .                          X                ∈        K        ⊆                              R                                m            ×            n                          ,            when the scalar functions   f  (      X    ) and   g  (      X    ) are nonnegative on   K, and   p  &amp;gt;  0.Is this true? If not, how to find a counterexample?

billyfcash5n · Accepted Answer

Take   f  (  x  )  =  (  2  −  x  ) and   g  (  x  )  =      x   for   x  ∈  [  0  ,  2  ]Take   p  =  2      h    1    (  x  )  =  f  +  g      h    2    (  x  )  =      f    2    +      g    2        h    1   is maximum for   x  =            1      4       and       h    2   is maximum for   x  =  0 (edited after comment...), with its minimum for             3      2

I was trying to solve the problem A maximization problem when I ask myself if the general problem

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