Calculus I · 2B · lesson
The Closed Interval Method
Learn the closed interval method with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Theorems, extrema, and curve shapeWhat this section is building
Learn the closed interval method with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Learning objectives
Apply the Extreme Value Theorem and closed interval method.
Absolute Extrema on a Closed Interval
Before the formulas
The theorem in The Closed Interval Method connects global information on an interval with local derivative behavior inside it. Read the hypotheses and conclusion separately. The theorem guarantees existence of at least one point; it may not identify the point, make it unique, or place it at the midpoint.
Use a diagram to understand the claim, then return to algebra to find candidate values when the problem asks for them. A correct theorem citation should name the interval and explain why each hypothesis is satisfied.
Absolute extrema are found by comparing a finite candidate list
On a closed interval, a continuous function must attain both an absolute maximum and an absolute minimum. Candidates occur at endpoints and critical numbers inside the interval. The method is therefore simple: find candidates, evaluate the original function, and compare outputs.
Do not compare derivative values. The derivative finds candidates; the original function decides which candidate is highest or lowest.
On a closed interval, absolute extrema may occur at interior critical numbers or at endpoints. The closed-interval method is therefore a finite comparison procedure: find every candidate, evaluate the original function, and compare the outputs.
This method is reliable because continuity on a closed bounded interval guarantees that absolute extrema exist. Without those hypotheses, a function may approach a best value without ever attaining it.
Extreme Value Theorem
If is continuous on a closed interval , then attains an absolute maximum and an absolute minimum on .
Closed interval method
• Find critical numbers inside . • Evaluate at every interior critical number. • Evaluate and . • Compare the function values. Largest is absolute maximum; smallest is absolute minimum.
Absolute extrema of a cubic
Find the absolute extrema of
on .
Worked solution
Write a real attempt before opening the supplied answer.
Endpoints do not need to be critical numbers to contain absolute extrema. Omitting endpoints from a closed-interval problem is a reliable way to calculate the wrong answer with impressive algebra.
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