I am reading a paper and they mention a hump function maximization: I am trying to prove the point o

rzfansubs87 2022-07-03 Answered
I am reading a paper and they mention a hump function maximization: I am trying to prove the point of maximization:
m = ( 1 x ) 1 σ x σ where x , σ [ 0 , 1 ]
It is said that m is a hump-shaped function of x maximized at x = σ, where σ is a parameter that is assumed to be fixed here.

My attempt:
First derivative with respect to x: ( 1 x ) σ x σ + ( 1 x ) 1 σ x σ 1 = 0
1 + ( 1 x ) 1 x 1 = 0
Second derivative with respect to x: ( 1 x ) σ 1 x σ + ( 1 x ) σ x σ 1 ( 1 x ) σ x σ 1 + ( 1 x ) 1 σ x σ 2
I couldn't get to the given result based on the above. Any clarification would be appreciated!
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Answers (1)

Gornil2
Answered 2022-07-04 Author has 20 answers
If x maximizes m then it will also maximize
log ( m ) = ( 1 σ ) log ( 1 x ) + σ log ( x )
because log is strictly monotonic increasing.
So let us try to find the maximum by setting 0 = m and trying to solve for x:
0 = ! log ( m ) x = ( 1 σ ) 1 x 1 + σ 1 x 0 = ( 1 σ ) x + σ ( x 1 ) = x σ x = σ
Now it should be easy to reason that m is maximal at x = σ.
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