Define the function f ( a , b , c , &#x03B1;<!-- α --> , &#x03B

Leland Morrow

Leland Morrow

Answered question

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

Beginner2022-06-25Added 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.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?