Suppose we have two functions, f ( x ) , g ( x ), and f ( x ) is convex and g ( x ) is concave. What are some conditions that will guarantee that f ( x ) g ( x ) is concave?

Question

Suppose we have two functions,   f  (  x  )  ,  g  (  x  ), and   f  (  x  ) is convex and   g  (  x  ) is concave. What are some conditions that will guarantee that   f  (  x  )  g  (  x  ) is concave?

lelestalis80d · Accepted Answer

Let&#039;s look at the smooth case.  (  f  g      )    ″    =      f    ″    g  +  2      f    ′        g    ′    +  f      g    ″  So you have three terms on the right, which are each positive or negative depending on   f and   g being positive or negative, increasing or decreasing, convex or concave.If you want a nice condition that makes   f  g concave, you&#039;ll want its second derivative   ≤  0, and it would be nice if each term on the right was   ≤  0: otherwise you&#039;d have to compare the sizes of the positive and negative terms, and things get messy. So if   f is convex you&#039;ll want   g  ≤  0 to make the first term   ≤  0; you&#039;ll want one of   f and   g nonincreasing and the other nondecreasing to make the second   ≤  0; and if   g is concave you&#039;ll want   f  ≥  0 to make the third term   ≤  0.

Suppose we have two functions, f(x),g(x) , and f(x) is convex and g(x) is concave. What are some conditions that will guarantee that f(x)g(x) is concave?

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