Extreme Value Theorem — Mathematics Knowledge System

Overview

The Extreme Value Theorem (EVT) guarantees that a continuous function on a closed, bounded interval [a,b] attains both a maximum and a minimum value somewhere on that interval. It's an existence theorem: it doesn't say how to find the extrema, only that they're guaranteed to exist, which is what makes optimization on closed intervals well-posed.

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Formal Definition

Definition

If f is continuous on the closed interval [a,b], then there exist points c, d in [a,b] such that:

f(c) \le f(x) \le f(d) \quad \text{for all } x \in [a,b]

f(c) is the absolute minimum and f(d) is the absolute maximum of f on [a,b]

Extreme Value Theorem

Worked Examples

f is a polynomial, hence continuous everywhere, so it's continuous on the closed interval [0,5] — EVT applies.
$f(x) = x^2 - 4x + 3$
Find critical points: f'(x) = 2x - 4 = 0 ⟹ x = 2 (in [0,5]).
$f'(x) = 2x-4$
Evaluate f at the critical point and both endpoints: f(0)=3, f(2)=-1, f(5)=8.
$f(0)=3,\ f(2)=-1,\ f(5)=8$

Answer: Absolute minimum is -1 at x=2; absolute maximum is 8 at x=5.

Practice Problems

Difficulty 3/10

Find the absolute maximum and minimum of f(x) = x³ - 3x on [-2, 2].

Difficulty 5/10

A simply-supported beam's bending moment is M(x) = 400x − 50x² (N·m) for 0 ≤ x ≤ 8 m. Find the location and value of the maximum bending moment.

Quiz

The Extreme Value Theorem guarantees a function attains an absolute max and min provided it is:

differentiable everywherecontinuous on a closed, bounded interval [a,b]increasinga polynomial

When optimizing a continuous function on [a,b], the absolute extremes can occur:

only at endpointsat critical points (f'=0 or undefined) OR at the endpointsonly where f'=0nowhere

Summary

The Extreme Value Theorem states a function continuous on a closed interval [a,b] must attain an absolute max and absolute min on that interval.
It is an existence theorem — it guarantees extrema exist without telling you where they are.
Both hypotheses matter: continuity and closedness of the interval. Drop either and the guarantee fails.
In practice, find extrema by checking f at critical points (where f'=0 or f' doesn't exist) and at the endpoints.

References

WebsiteWikipedia — Extreme value theorem