5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

Extreme Value Theorem

If a function f is continuous on a closed interval [a, b], then f must have both an absolute (global) maximum and an absolute (global) minimum on that interval. These extrema can occur at endpoints or at interior points. Continuity and closedness are both required — without them, extrema may not exist.

Critical Points

A critical point occurs where f'(x) = 0 or f'(x) is undefined (and x is in the domain of f). Critical points are the only candidates for local extrema inside an interval — but not every critical point is an extremum (e.g., a saddle-like point where f' doesn't change sign).

Local vs. Global Extrema

A local (relative) extremum is the highest or lowest value in some neighborhood of a point, while a global (absolute) extremum is the highest or lowest over the entire domain/interval. To find global extrema on [a, b], evaluate f at all critical points and both endpoints, then compare values.

Key terms

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a, b], then f has both an absolute maximum and an absolute minimum on that interval.

Critical point

A point c in the domain of f where f'(c) = 0 or f'(c) is undefined.

Local (relative) extremum

The highest or lowest value of f in a neighborhood around a point, not necessarily over the entire domain.

Global (absolute) extremum

The highest or lowest value of f over its entire domain or given interval.

How to find absolute extrema on [a, b]

Evaluate f at all critical points in (a, b) and at both endpoints a and b, then compare the values.

Try an AP-style question

What are the critical points of f(x) = x³ − 3x?

1. x = 3 only
2. x = −1 and x = 1
3. x = −√3, 0, and √3
4. x = 0 only