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Extreme Value Theorem

The Extreme Value Theorem (EVT) is one of the most important existence results in analysis: if $f$ is continuous and the domain $D$ is compact (closed and bounded), then $f$ attains its maximum and minimum values on $D$ .

More precisely, there exist points $\mathbf{a}^*, \mathbf{b}^* \in D$ such that $f(\mathbf{a}^*) \leq f(\mathbf{x}) \leq f(\mathbf{b}^*)$ for all $\mathbf{x} \in D$ .

The EVT guarantees existence but does not tell us how to find these extrema. Practically, we check: (1) interior critical points (from $\nabla f = \mathbf{0}$ ), (2) boundary critical points, and (3) corners/vertices if the domain is a polytope. The global extremum is the best of all these candidates.

Formal View

Theorem 12.12 (Extreme Value Theorem) — Existence of Global Extrema

f: D \to \mathbb{R}

is continuous and

D \subseteq \mathbb{R}^n

is compact, then

f

attains its minimum and maximum on

D

: there exist

\mathbf{a}^*, \mathbf{b}^* \in D

such that

f(\mathbf{a}^*) = \min_{\mathbf{x}\in D} f(\mathbf{x})

and

f(\mathbf{b}^*) = \max_{\mathbf{x}\in D} f(\mathbf{x})

Interactive Visualization

Interactive Line Explorer

Why This Matters

The EVT is the theoretical foundation for asserting that optimization problems on compact domains have solutions.

Guaranteeing that constrained optimization problems have optimal solutions
Existence of optimal portfolios under budget constraints
Proving that differential equations have solutions under regularity conditions

Learning Resources

Extreme Value Theorem

Khan Academy

Statement and intuition of the extreme value theorem.

8 min

EVT in Multiple Variables

MIT OpenCourseWare

Applying the extreme value theorem to constrained optimization.

48 min

Quiz

Question 1

The Extreme Value Theorem requires which conditions?

Question 2

The Extreme Value Theorem tells us how to find the minimum — not just that it exists.

Common Mistakes

Trying to apply the EVT to continuous functions on non-compact (e.g., open or unbounded) domains.
Confusing existence (EVT) with uniqueness — the EVT guarantees a minimum exists but does not say it is unique.
Forgetting that the minimum might be attained at the boundary, not just at interior critical points.