Definition 2.4.1. Let $f:D\to \mathbb R$ be a function and let $c\in D$. If
\[\forall x\in D\ f(c)\ge f(x)\]then $f(c)$ is called the global maximum value of $f$. Or, if
\[\forall x\in D\ f(c)\le f(x)\]then $f(c)$ is called the global minimum value of $f$. And the global maximum and minimum values of $f$ are called extreme values of $f$.
Definition 2.4.2. Let $f:D\to \mathbb R$ be a function and let $c\in D$. If there exists an open interval $I\subseteq D$ such that
\[c\in I \qquad \text{and} \qquad \forall x\in I\ f(c)\ge f(x)\]then $f(c)$ is called a local maximum value of $f$. Or, if there exists an open interval $I\subseteq D$ such that
\[c\in I \qquad \text{and} \qquad \forall x\in I\ f(c)\le f(x)\]then $f(c)$ is called a local minimum value of $f$. And local maximum and minimum values of $f$ are called local extreme values of $f$.
Theorem 2.4.3 (The Extreme Value Theorem). Let $f:D\to \mathbb R$ be a function. If $f$ is continuous on a closed interval $I$, then $f$ attains both a global maximum value and a global minimum value in $I$.
Theorem 2.3.4 (Fermat’s Theorem). Let $f:D\to \mathbb R$ be a function and let $c\in D$. If $f$ has a local maximum or minimum at $c$ and $f$ is differentiable at $c$, then
\[f'(c)=0.\]Proof.