The Closed Interval Method

Definition

The Closed Interval Method is a systematic procedure for finding the absolute maximum and minimum values of a continuous function $f$ on a closed interval $[a, b]$. It is the practical application of the Extreme Value Theorem.

Why It Matters

It provides a guaranteed way to find the absolute maximum and minimum of a system, which is critical for optimization problems where boundary conditions are paramount.

Core Concepts

The method consists of three steps:

1. Find Critical Numbers: Calculate $f'(x)$ and identify all critical numbers $c$ within the open interval $(a, b)$ where $f'(c) = 0$ or $f'(c)$ does not exist.
   • How to read: “The derivative f prime of c equals zero, or f prime of c does not exist.”
   • Meaning: Critical points are where the slope vanishes or is undefined—candidate locations for interior extrema.
2. Evaluate at Candidates: Calculate the value of the function $f$ at:
   • Each critical number found in step 1.
   • The endpoints of the interval, $a$ and $b$.
3. Compare Results: The largest of these values is the absolute maximum; the smallest is the absolute minimum.

Connected Concepts

• The Extreme Value Theorem
• Critical Points
• Absolute Extreme Values
• Extrema
• Applied Optimization Strategy
• Bottlenecks
• Leverage
• Trade-offs
• Scale