5.2 Extreme Value Theorem & Critical Points

Extreme Value Theorem

A function continuous on a closed interval [a, b] attains both an absolute maximum and an absolute minimum on that interval.

Critical points

A critical point is where f'(x) = 0 or f'(x) is undefined. Local (relative) extrema occur only at critical points; absolute extrema on [a, b] occur at critical points or at the endpoints.

Key terms

Extreme Value Theorem

Continuous f on [a, b] attains an absolute max and min.

Critical point

Where f'(x) = 0 or is undefined.

Local extrema

Occur only at critical points.

Absolute extrema

At critical points or endpoints of [a, b].

EVT requirement

Closed interval and continuity.

Try an AP-style question

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

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