Interior Extremum

When you sketch the graph of a function and mark the “peaks” and “valleys”, you are locating its local extrema. Before you can prove any theorem about them — such as Fermat’s Lemma — you need precise definitions that distinguish local from global, interior from boundary, and strict from non-strict.

Local maximum and minimum

Let $f$ be defined on a set $D \subseteq \mathbb{R}$ and let $x_0 \in D$.

Definition. $x_0$ is a local maximum of $f$ if there exists $\delta > 0$ such that

$$f(x) \leq f(x_0) \quad \text{for all } x \in D \text{ with } |x - x_0| < \delta.$$

If the inequality is strict ($f(x) < f(x_0)$ for $x \neq x_0$), $x_0$ is a strict local maximum.

A local minimum and strict local minimum are defined symmetrically by reversing the inequality. The term local extremum covers both cases.

Interior vs. boundary extrema

The definitions above apply equally to interior points and boundary points of $D$. However, most theorems about extrema — including Fermat’s Lemma — require $x_0$ to be in the interior of $D$, meaning there exists $\delta > 0$ such that $(x_0 - \delta, x_0 + \delta) \subseteq D$.

Why the distinction matters. At a boundary point, the function is only compared with values on one side. For example, $f(x) = x$ on $[0, 1]$ has a local minimum at $x_0 = 0$ and a local maximum at $x_0 = 1$, even though $f' = 1 \neq 0$ at both endpoints. A stationarity condition like $f'(x_0) = 0$ can only hold at interior points.

Local vs. global extrema

A global maximum (also called an absolute maximum) is a point $x_0 \in D$ where $f(x_0) \geq f(x)$ for all $x \in D$ — not just nearby. Every global extremum is a local extremum, but not conversely.

Example. For $f(x) = \sin x$ on $\mathbb{R}$: every point where $\sin x = 1$ is both a local and a global maximum, while points where $\sin x = -1$ are both local and global minima. There is no other local extremum because $\sin x$ oscillates to arbitrarily large values of $x$ in each direction.

Example. For $f(x) = x^3 - 3x$ on $\mathbb{R}$: $x = -1$ is a local maximum with $f(-1) = 2$, and $x = 1$ is a local minimum with $f(1) = -2$. Neither is a global extremum since $f(x) \to \pm\infty$ as $x \to \pm\infty$.

Characterisation via local properties

A point $x_0 \in \operatorname{int}(D)$ is a local maximum of $f$ if and only if $f - f(x_0)$ is non-positive in some neighbourhood of $x_0$. This reformulation connects directly to the sign-analysis of the difference quotient used in Fermat’s Lemma.

Summary

• $x_0$ is a local maximum if $f(x) \leq f(x_0)$ for all $x$ near $x_0$ (within $D$); strict if the inequality is sharp for $x \neq x_0$.
• Interior extrema lie in the open interior of the domain; boundary extrema do not.
• Every global extremum is a local extremum; the converse fails.
• Theorems that derive conditions from $f'(x_0) = 0$ require the extremum to be interior and $f$ to be differentiable there.