Lesson 2: 5.2 Extreme Value Theorem and Critical Points
Duration of Days: 2
Lesson Objective
Identify critical points of a function by finding where f'(x) = 0 or where f'(x) is undefined.
Distinguish between relative (local) extrema and absolute (global) extrema.
Apply the Extreme Value Theorem (EVT): understand that if a function is continuous on a closed interval [a, b], it must have both an absolute maximum and an absolute minimum.
Perform the Candidates Test to find absolute extrema by comparing function values at critical points and endpoints.
Can a function have a maximum value if it is not continuous?
Why must we check the endpoints of an interval when looking for the absolute maximum?
Is every critical point a relative maximum or minimum?
How does the derivative help us find the "peaks and valleys" of a function without looking at a graph?
Extreme Value Theorem (EVT)
Critical Point (Critical Value)
Absolute (Global) Maximum/Minimum
Relative (Local) Maximum/Minimum
Closed Interval
Extremum (plural: Extrema)
Candidates Test
Standard FUN-1.C: Explain the relationship between critical points and the locations of relative extrema.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Description:The lesson introduces the "Critical Point" as the primary suspect for a maximum or minimum. Students learn the procedural Candidates Test:
Find f'(x).
Set f'(x) = 0 and find where f'(x) is undefined to find critical points.
Plug the critical points and the endpoints (a and b) into the original function f(x).
The largest result is the absolute max; the smallest is the absolute min.
Purpose: The EVT is a cornerstone of AP Calculus. The Candidates Test is a required justification on many Free Response Questions. Students often forget to check endpoints, so this lesson builds the habit of "Checking the boundaries."
DOK 1 (Recall): Defining a critical point and stating the requirements for the EVT.
DOK 2 (Skill/Concept): Finding critical points for polynomial, rational, and radical functions.
DOK 3 (Strategic Thinking): Determining the absolute extrema on a closed interval and justifying why a specific point is the maximum using the Candidates Test.
Scaffolding (Support):The "Candidates List" Table: Provide a table with three columns:
"Candidate (x-value),"
"Source (Endpoint or CP),"
and "Function Value f(x).
"Visual Sorting: Give students a graph with several points marked and have them categorize each as "Local Max," "Global Max," "Endpoint," etc.
Extension (Challenge):Open Intervals: Ask students what happens to the EVT if the interval is open (a, b) instead of closed [a, b].
Can they find a function that has no maximum on an open interval?
Piecewise Functions: Challenge students to find the absolute extrema of a piecewise function where the "break" in the function occurs at a critical point.
AP College Board Assessments