Extreme Value Theorem

111. Extreme Value Theorem

The Extreme Value Theorem states that if a function $𝑓 (𝑥)$ is continuous on a closed interval $[𝑎, 𝑏]$ , then $𝑓 (𝑥)$ must attain both a maximum and a minimum value within that interval. This means there exist points $𝑐, 𝑑 \in [𝑎, 𝑏]$ such that:

𝑓 (𝑐) \geq 𝑓 (𝑥) and 𝑓 (𝑑) \leq 𝑓 (𝑥) for all 𝑥 \in [𝑎, 𝑏]

Continuity: The function must be continuous on $[𝑎, 𝑏]$ . Discontinuities (jumps, asymptotes, holes) can prevent the function from attaining an extreme value

Example

Closed Interval: If the function is defined on an open interval $(𝑎, 𝑏)$ , an extremum may not exist

Example

$𝑓 (𝑥) = \frac{1}{𝑥}$ on $(0, 1]$ has no maximum because it keeps increasing as $𝑥 \to 0$ .