I'm having trouble with this real analysis/optimization problem. The result seems intuitively obvious, but I don't know how I could possibly formalize it. Any advice is appreciated:
Let $X \subset \mathbb{R}^n$, $Y \subset \mathbb{R}$, $f:x \rightarrow Y$ and $g:: Y \rightarrow \mathbb{R}$.
Suppose $X$ is a compact set, $f$ and $g$ are continuous functions on their domains, and $g$ is a strictly monotonically decreasing function on its domain.
Argue that the sets referred to exist and are equivalent:
$\text{argmax}_{\textbf{x} \in X} f(\textbf{x}) = \text{argmin}_{\textbf{x} \in X} g \circ f(\textbf{x})$