Local extrema

Local extrema are either local minima or local maxima. Every extremum is a Critical point, but not the converse.

Put simply, a point is a local minimum of a function $𝑓$ iff. all points in the immediate neighbourhood are greater, and vice versa for maxima. More formally, we say $𝑐$ is a local minimum iff.

\exists 𝜖 \in (0, \infty) . \forall 𝑥 \in (𝑐 - 𝜖, 𝑐 + 𝜖) . 𝑓 (𝑥) > 𝑓 (𝑐)

and a maximum iff.

\exists 𝜖 \in (0, \infty) . \forall 𝑥 \in (𝑐 - 𝜖, 𝑐 + 𝜖) . 𝑓 (𝑥) < 𝑓 (𝑐)

The absolute extrema are guaranteed to exist for certain functions and domains — see Extreme Value Theorem.

First derivative test

For any critical point $𝑐$ , $𝑓' (𝑐) = 0$ . See Critical point on classifying critical points.

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