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

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Cullen Petersen · Accepted Answer

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        &amp;gt;  0

inurbandojoa · Accepted Answer

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&#039;ll also need some restrictions to       f    ″    ,      g    ″   as well as   f  ,  g for the proof to work.

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

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