Given the subsets $M \subseteq \mathbb{R}$ and mappings $f : M \to \mathbb{R}$, determine whether $\max(M, f)$ and $\min(M, f)$ exist and provide your justifications.$M = \{(x, y)^{t} \in \mathbb{R}^2 : 1 \leq x^2 + y^4 \leq 2\}$, $f(x, y) = e^{xy} - \cos(\sin(x^2))$
$\textbf{Solution:}$ To determine the existence of $\max(M, f)$ and $\min(M, f)$, we need to analyze the function $f$ over the subset $M$.
First, let's consider the lower bound of $f$ over $M$. Since $f$ is continuous and $M$ is a closed and bounded subset of $\mathbb{R}^2$, by the Extreme Value Theorem, $f$ attains its minimum value over $M$. Therefore, $\min(M, f)$ exists.
Next, we consider the upper bound of $f$ over $M$. We observe that $f(x, y)$ is unbounded as $(x, y)$ approaches the boundary of $M$. Specifically, as $x^2 + y^4$ approaches $1$ or $2$, the value of $f(x, y)$ increases without bound. Therefore, $\max(M, f)$ does not exist.
Is my solution detailed enough?