Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex.
If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, will this new function be convex or concave?
I was wondering is there is an easy way to determine this, instead of deriving it from scratch, (like maybe there is a property or rules relating to operations on con-vex/cave functions or something). Thanks.
Best Answer
There is no general rule. Let $f(x)=1$; then $f$ is both concave and convex. Let $g(x)=e^{x^2}$, which is convex. However $$ \frac{f(x)}{g(x)}=e^{-x^2} $$ is neither concave nor convex.