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Research draft

maxima and minima

vr.tr.maxima-and-minima · XCT.QLT

Let an agent explain maxima and minima and how they are found, distinguish global, local and boundary extrema, support solving optimisation problems, and describe applications across fields.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain maxima and minima and how they are found, distinguish global, local and boundary extrema, support solving optimisation problems, and describe applications across fields.

The largest and smallest values of a function or set, either globally over the whole domain or locally in a neighbourhood, found in calculus at critical points where the derivative vanishes or fails to exist and at boundaries, and in ordered sets as maximal and minimal elements; maxima and minima are the subject of optimisation, appear in physics, economics, engineering and geography such as the northernmost point of a region, and are computed analytically and numerically.

What it is for: Largest and smallest values.

It can be explain extrema; distinguish global and local; support optimisation problems; describe applications.

Distinguishing features

Extreme values

Critical points

Global versus local

Optimisation

What it looks like

Not physical; points on graphs or elements of sets.

How it is recognised

Highest and lowest values

Global or local

Saddle points are critical but not extrema

Related models

is a kind of - category

critical point

is related to - second derivative tests

curvature

is related to - cost minimisation

unit cost

is related to - extreme points of regions

tripoint

In practice

Families and kinds

global maxima and minima

local extrema

boundary extrema

maximal and minimal elements in orders

constrained extrema

Standards and regulation

Mathematical definitions and notation

Numerical optimisation software conventions

Failure modes and hazards

Confusing local and global extrema

Ignoring boundaries and non-differentiable points

Numerical methods trapped in local optima

Also called

coordinate of northernmost pointpricelessnesscomassmaximumminimummaximal elementmaxima of a point setrated maximumRAM limitruling gradientmaximum profit at maturitymaximum loss at maturityrated minimumminimal prime idealwetting currentglobal minimumlocal minimummaximal and minimal elementsarg max and minlocal extremumglobal extremum

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Understand What extrema are.

Mathematics.

Definition

Definitions.

Definition

Definitions.

  1. What are global and local maxima and minima, and how do they relate to critical points and boundaries? definition
  2. What are maximal and minimal elements in partially ordered sets? definition

Tests

Finding extrema.

Tests

Tests.

  1. How are extrema found using first and second derivative tests, and in several variables using gradients and Hessians? definition
  2. Which entry fits calculus? action
Solve Solving problems.

Practice.

Calculate

Computing extrema.

Calculate

Computing.

  1. What are the maxima and minima of this function on this domain? action
  2. Have boundaries and non-differentiable points been checked? boundary

Constrained

Constrained optimisation.

Constrained

Constrained.

  1. How are extrema found under constraints using Lagrange multipliers or numerical methods? action
  2. Which entry fits optimisation? action
Apply Applications.

Context.

Fields

Extrema in fields.

Fields

Fields.

  1. How do physics, economics and engineering use extrema, from least action to cost minimisation? provenance
  2. Which entry fits a specific application? action

Geography

Extreme points.

Geography

Geography.

  1. How are extreme points such as the northernmost point of a country determined? provenance
  2. Which entry fits extreme points? action
Learn History and teaching.

Education.

History

History.

History

History.

  1. How did the theory of extrema develop from Fermat to modern optimisation? provenance
  2. Which references are standard? provenance

Teach

Teaching.

Teach

Teaching.

  1. How can extrema be taught with graphs and applied problems? action
  2. Which misconceptions arise? provenance

What the second pass must settle

  • Should optimisation be a separate entry?
  • How should textbooks be linked?
  • How should software be linked?
Draft as JSON