---
tag: 5X1YMSR204
kind: concept
name: Maximum and minimum
msc: [26A06, 49K05]
sources:
  - url: https://en.wikipedia.org/wiki/Maximum_and_minimum
same_as:
  wikidata: Q845060
  wikipedia: Maximum and minimum
---

## Summary

The maximum of a real function on a set is its largest value, and the minimum is its smallest value.
A point where the maximum or the minimum is attained is a global extremum; this is stronger than a [[local extremum|0WFBH1Q3WC]].
A function need not attain a maximum, but by the [[extreme value theorem|0WS1A8E2P9]] a continuous function on a closed bounded interval does.
Finding maxima and minima is the aim of optimization, and derivatives are the main tool.

## Explanations

### L1 human

The maximum of a function is the highest point of its graph, and the minimum is the lowest point.
On $[-1, 2]$, the function $x^2$ has its maximum $4$ at $x = 2$ and its minimum $0$ at $x = 0$.
Some functions have no highest point: on the open interval $(0, 1)$, the function $x$ gets close to $1$ but never reaches it.