Question 1027568
Let {{{df/dx < 0}}}

A)  {{{h(x) = (f(x))^2}}} ==> h'(x) = 2f(x)f'(x) < 0, because f(x) > 0 always and f'(x) < for all real numbers x.
Thus h(x) is decreasing for all real numbers x.  Nowhere will it be increasing.


B) j(x) = f(f(x)) = (fof)(x) ==> j'(x) = {{{(df(f(x))/df(x))*(df(x)/dx)}}}.
Now the domain of fof is a subset of the domain of f(x), hence by hypothesis,  {{{df(f(x))/df(x) < 0}}}.  Also {{{df(x)/dx<0}}} again by hypothesis.
==> j'(x) > 0, and so j(x) will always be increasing in the domain of (fof)(x).