Decreasing exponential

Consider the function $$f(x) = A e^{xB} e^{-Ce^x} (1 + O(e^{-x})),$$ where $ A, B, C \in \mathbb{R} $, under what conditions on $A, B,$ and $C$ will $f(x)$ be decreasing as $x$ goes large? How do the signs and magnitudes of $B$ and $C$ influence the behavior of $f(x)$ at infinity, and what role does the small $O(e^{-x})$ term play in this context? Specifically, what is the relationship between $B$ and $C$ that ensures the function's derivative $f'(x)$ is negative for sufficiently large $x$?

Any help or hint's would be so much appreciated!