Extrema

A local minimum of a Scalar Field $𝑓$ is a value $𝑓 (𝑎)$ such that

𝑓 (𝑎) \leq 𝑓 (𝑥)

Similarly, a local maximum of a scalar field $𝑓$ is a value $𝑓 (𝑎)$ such that

𝑓 (𝑎) \geq 𝑓 (𝑥)

Where the domain is an open set containing

The local extrema of a Scalar Field $𝑓$ can be determined by

𝑓 (𝑎) \leq 𝑓 (𝑥)

𝑓 (𝑎) \geq 𝑓 (𝑥)

For all positions $𝑥 \in 𝐷$

A critical point of a function $𝑓$ is a position $𝑎$ such that $𝑓$ is not differentiable

\nabla 𝑓 (𝑎) = 0

If $𝑓 (𝑎)$ is a local extremum, then $𝑎$ is a critical point