1. A real valued function $f$ is said to increasing if and only if for all $x$, $y$ in the domain of $f$ such that $x < y$, we
have $f(x) \leq f(y)$. $f$ is said to be strictly increasing if the latter inequality holds strictly.
(a) Prove the following theorem:
THEOREM Let $S \subset \mathbb{R}^n$ and $f: S \to \mathbb{R}$. Let $F: \mathbb{R} \to \mathbb{R}$ be a function defined over the range of $f$, and
let $c \in S$. Consider the function $g(x) = F \circ f(x) = F(f(x))$.
(i) If $F$ is increasing and $c$ maximizes [minimizes] $f$ over $S$, then $c$ also maximizes [minimizes] $g$ over $S$.
(ii) If $F$ is strictly increasing, then $c$ maximizes [minimizes] $f$ over $S$ if and only if $c$ maximizes [minimizes]
g over $S$.
This theorem states that maximizing a function is equivalent to maximizing a strictly increasing transformation of that function.
Now, consider $f(x) = e^{-x^2 - y^2}$.
(b) Let $F(u) = \ln u$. Verify that the two functions $f$ and $F \circ f$ both have maxima at the same values of $(x, y)$.
(c) Let $F(u) = 5$. Explain why this example shows that the implication in part (a) cannot be reversed.
