Extrema

Definition

Extrema are the maximum and minimum values that a function attains over its domain (global) or within a local interval (local).

Why It Matters

Optimization is the search for extrema; whether you are seeking the maximum possible profit or the minimum fuel consumption, identifying and distinguishing peaks is key to making efficient decisions.

Core Concepts

• Absolute vs. Relative Extrema:
  • A Global (Absolute) Maximum is the largest value a function attains over its entire domain.
  • A Local (Relative) Maximum is the largest value a function attains within a small neighborhood.
• The Extreme Value Theorem (EVT): If a function $f$ is continuous on a closed interval $[a, b]$, then $f$ must attain an absolute maximum and an absolute minimum at least once.
  • How to read: “The function f is continuous on the closed interval from a to b.”
  • Meaning: Continuous functions on closed bounded intervals are guaranteed to have absolute extrema.
• Fermat’s Theorem: If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.
  • How to read: “The existence of a local maximum or minimum at c, where f prime of c exists, implies that f prime of c is equal to zero.”
  • Meaning: At a smooth turning point, the tangent is horizontal — slope must be zero.
• Critical Numbers: A value $c$ is a critical number if $f'(c) = 0$ or $f'(c)$ does not exist. All local extrema occur at critical numbers.
  • How to read: “The value f prime of c is equal to zero, or f prime of c does not exist.”
  • Meaning: Candidate points for local extrema.

Connected Concepts

• The First Derivative Test for Local Extrema
• The Second Derivative Test for Local Extrema
• Critical Points
• The Extreme Value Theorem
• Applied Optimization Strategy