Define the function f ( a , b , c , α , β , γ , x ) = max ( 0 , max ( ( a + x ) α , ( b + x ) β ) − ( c + x ) γ ) ,where a , b , c , α , β , γ , x ∈ [ 0 , M ] .Is it true that for any ξ = ( a , b , c , α , β , γ ) ,the maximum of f ( ξ , ⋅ ) occurs either when x = 0 or x = M?I think the answer is yes, but I have trouble prooving it.My argument is as follows:Given any ξ, f ( ξ , c ) will be f ( ξ , x ) = { 0 case A ( a + x ) α − ( c + x ) γ case B ( b + x ) β − ( c + x ) γ case C Hence, f ( ξ , x ) is a linear function of x in all three cases, and the result follows.I think this argument works only if each case is independent of x, but this is not the case.As, when ξ is given, judiciously choosing x may put f in another case.What would be a right way to prove this result?

Question

Define the function  f  (  a  ,  b  ,  c  ,  α  ,  β  ,  γ  ,  x  )  =  max              (        0  ,    max              (            (    a    +    x    )    α  ,      (    b    +    x    )    β            )        −      (    c    +    x    )    γ            )        ,where  a  ,  b  ,  c  ,  α  ,  β  ,  γ  ,  x  ∈      [    0    ,    M    ]    .Is it true that for any  ξ  =      (    a    ,    b    ,    c    ,    α    ,    β    ,    γ    )    ,the maximum of   f  (  ξ  ,  ⋅  ) occurs either when   x  =  0 or   x  =  M?I think the answer is yes, but I have trouble prooving it.My argument is as follows:Given any   ξ,   f  (  ξ  ,  c  ) will be  f  (  ξ  ,  x  )  =      {                            0                          case A                                      (          a          +          x          )          α          −          (          c          +          x          )          γ                          case B                                      (          b          +          x          )          β          −          (          c          +          x          )          γ                          case C                        Hence,   f  (  ξ  ,  x  ) is a linear function of   x in all three cases, and the result follows.I think this argument works only if each case is independent of   x, but this is not the case.As, when   ξ is given, judiciously choosing   x may put   f in another case.What would be a right way to prove this result?

Leland Ochoa · Accepted Answer

It appears that   f is a concatenation of convex functions (namely &quot;max&quot; and linear operations), and therefore   f is convex. It follows that one of the end points will always be maximal.

Define the function f ( a , b , c , α , &#x03B

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