Find function that maximize a integral

Question

Let $x\in [0,1]$ and arbitrary positive function $f(x)>C>0$. My question is how to find a function $A(x)\in [0,1]$ in order to maximize the integral below: $$\int_{0}^{1} A(x)\,d\,ln\,f(x)$$

Any direction or thoughts (e.g. adding more constrains to $A(x)$) is greatly appreciated.

Answer 1

I just did more research on the topic and thinking this is related to a more general solution named as Euler-Lagrange equation. Basically $A(x)$ can possibly be found by searching the stationary points of its functional. I will provide more constructive answer if I can think through this.