Optimizing a composition of functions

Suppose I want to optimize `f(x)` (i.e., find the arg min/max: the x which minimizes / maximizes the function). Calc I taught me that I can simply find `df(x)/dx` and set it to zero, then solve for x.

Suppose that's hard.

Under what conditions can I instead optimize `g(f(x))`? What does `g()` have to satisfy?

What I know:

* The minimizer should be the same, since `d g(f(x))/dx=0` is `g'(f(x)) * f(x)=0` is the same. But what if `g'(f(y))=0` but `f'(y) is not 0`? Then we've introduced a "false" root, correct?

EDIT: Sorry for switching notation. `g'` is the first derivative of `g` with respect to `x`.