Need some help on this one and I thought this was probably the best place to turn. Here's the problem:
Suppose f & g are both concave upward on (-infinity,infinity). Under what condition on f will the composite function h(x)= f(g(x)) be concave upward?
Any help would greatly be appreciated.
Thanks TroyHi Troy,
While it would be appealing to 'see' this in some simple visual way, I do not 'see' how, yet.
Working with algebra and calculations:
concave up = second derivative positive.
[f(g)]" = [f'(g).g']' = f"(g).g'2 + f'(g).g"
So, the question of whether this SUM is positive is a bit obscure. The FIRST term f"(g).g'2 IS positive, since f" is assumed positive (concave up) and any square is positive OR zero.
The second term is positive IF f'(g) >0 (or =0) - i.e. if g(x) is in a section of f which is increasing (or non-decreasing).
Otherwise, one must know the balance of these two terms - which is bigger, etc.
I think, at this point, you can probably give some SUFFICIENT conditions, conditions under which you know for sure that f(g) is concave up.
This would probably NOT cover ALL POSSIBLE situations, but only some which are easily detected.
Still, it would be nice to have a simple visual image - but it does seem tricky.