Critical points/values and regular points/values

Let $𝑓 : 𝑋 \to 𝑌$ be a $𝐶 \infty$ differentiable map between $𝐶 \infty$ differentiable manifolds $𝑋, 𝑌$ of dimensions $𝑛, 𝑚$ respectively. A point $𝑥 \in 𝑋$ is called a critical point iff $r a n k 𝑇 𝑥 𝑓 < 𝑚$ where $𝑇 𝑥 𝑓$ is the Tangent map at $𝑥$ ; otherwise $𝑥$ is a regular point (and $𝑓$ is submersive at $𝑥$ ). The image of a critical point is a critical value, and a value which is not critical is a regular value.

Special cases

Single variable function

A critical point is either a Local extremum or a point of inflection (POI). It is defined as a point where the derivative is either 0 or undefined. #to/generalize

Classifying critical points

First derivative test

Given $𝑓' (𝑐) = 0$ ,

Iff. $\forall 𝑥 < 𝑐 . 𝑓' (𝑥) \geq 0$ and $\forall 𝑥 > 𝑐 . 𝑓' (𝑥) \leq 0$ then $𝑐$ is a Local maximum.
Iff. $\forall 𝑥 < 𝑐 . 𝑓' (𝑥) \leq 0$ and $\forall 𝑥 > 𝑐 . 𝑓' (𝑥) \geq 0$ then $𝑐$ is a Local minimum.
Otherwise, it is neither.

Second derivative test

We can also use the second derivative to test for Concavity at a critical point, which can classify a critical point. Given $𝑓' (𝑐) = 0$ ,

Iff. $𝑓 ″ (𝑐) < 0$ the critical point is Concave down and $𝑐$ is a Local maximum.
Iff. $𝑓 ″ (𝑐) > 0$ the critical point is Concave up and $𝑐$ is a Local minimum.

#state/tidy | #SemBr