Question

Maximize the function instead of composite function

Answer 1

here is the answer

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.