Basically, I am wondering when the derivative of a product will be "nice" in the sense that if each term has derivatives that are monotonic than the product will have a derivative that is monotonic.
Using the product rule, $$ (g(x)*h(x))' = g(x)h'(x) + g'(x)h(x) $$
If $g,h$ are both positive and non-increasing (non-decreasing) then it is clear that $(g(x)*h(x))'$ will be non-increasing (becuase each term will be negative).
I'm sure a similar logic will hold if both $g,h$ are negative.
if $g,h$ are og opposite signs, though, or if their derivatives are of opposite signs, then we cannot sign the terms, and so I think the product can be non-monotonic?
- If possible, can someone provide an example where $g'(x)$ and $h'(x)$ are monotonic, but $(g(x)*h(x))'$ is really non-monotonic (by really, I mean it switch from increasing to decreasing and back more than once)
Let $f(x)= x\,\forall \,x\in R$ and $ g(x)=(x-2)\,\forall\,x\in R$ $$ $$ $f(x)$ and $g(x)$ both are monotonic $\forall\,x\in R$ $$. $$ But $h(x)=f(x)g(x)=x(x-2)=(x-1)^2-1$. Is decreasing for $x\lt 1$. And increasing for $x\gt1$.