How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

pin1ta4r3k7b

pin1ta4r3k7b

Answered question

2022-11-19

How to prove that the product of a decreasing monotonic function and a strictly increasing monotonic function is a concave function?

Answer & Explanation

Cullen Petersen

Cullen Petersen

Beginner2022-11-20Added 13 answers

This is generally wrong if there are no other restrictions for your functions, e.g. when f ( x ) = e x 2 and g ( x ) = e x ,, then f is strictly increasing monotonic, g is decreasing monotonic, but h ( x ) is strictly convex on ( 0 , 1 ):
h ( x ) = 2 e x 2 x + ( 2 x 1 ) 2 e x 2 x > 0
inurbandojoa

inurbandojoa

Beginner2022-11-21Added 11 answers

There are always many ways to prove a statement. However the most simple (and elegant) one will be using the chain rule:
h ( x ) = ( f g ) ( x ) = ( f g + f g ) ( x ) = f g ( x ) + 2 f g ( x ) + f g ( x )
Here we see you'll also need some restrictions to f , g as well as f , g for the proof to work.

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