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Extreme Value Theorem

Theorem

Let a<ba<b be real numbers, and let f ⁣:[a,b]→Rf\colon [a, b] \to\mathbb{R} be a continuous function on [a,b].[a, b]\text{.} Then ff attains its global maximum at x=xmax∈[a,b],x=x_{\mathrm{max}} \in [a, b]\text{,} and also attains its global minimum at x=xmin∈[a,b].x=x_{\mathrm{min}} \in [a, b]\text{.}

Interpretation

The extreme value theorem guarantees that a continuous function on a closed and bounded interval attains both a largest and a smallest value. It does not say where they are, only that a search for them cannot come back empty.

A continuous function on a closed interval from a to b with its maximum attained inside the interval and its minimum attained at the endpoint b.

In the figure the maximum is attained at the interior point cc, the highest point of the graph, and the minimum at the endpoint bb. Both are values that ff actually takes, f(c)f(c) and f(b)f(b), not merely bounds that it approaches. The theorem allows either kind of location, so the endpoints always have to be checked.

Each hypothesis is needed. On the open interval (0,1)(0,1) the function f(x)=xf(x)=x comes arbitrarily close to 11 without reaching it, so it has no maximum; on the unbounded interval [0,+∞)[0,+\infty) the same function is not even bounded above. Without continuity a jump can remove the extreme value: on [0,1][0,1], the function equal to xx for x<1x<1 and to 00 at x=1x=1 has supremum 11 but never takes the value 11. The hypotheses are sufficient, not necessary: sin⁡x\sin x attains its maximum 11 on the open interval (0,π)(0,\pi), at x=π2x=\tfrac{\pi}{2}.

In the language of global extrema, the theorem says that a continuous ff on [a,b][a,b] has a global maximum and a global minimum; in particular, ff is bounded there and its supremum is a maximum. Math III extends the theorem from closed bounded intervals to a wider class of sets in the extreme value theorem on compact sets.

Key Results

  • The theorem does not apply to open or unbounded intervals.
  • The theorem does not apply to functions that are not continuous.

Examples

  • The function f(x)=x2f(x)=x^2 on [0,1][0,1] attains the largest value 11 and the smallest value 0.0\text{.}
  • The function f(x)=xf(x)=x has no maximum on (0,1).(0,1)\text{.}

Prerequisites

Sources

Aliases: EVT, Weierstrass extreme value theorem Symbols: element of, infinity, pi, real numbers

Last updated: 20 October 2026 03:02