Define the function f ( a , b , c , α , β ,...

Leland Morrow

Leland Morrow

Answered

2022-06-24

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?

Answer & Explanation

Leland Ochoa

Leland Ochoa

Expert

2022-06-25Added 25 answers

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

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