Justify conclusions about functions by applying the Extreme Value Theorem.
Distinguish between absolute and relative extrema and critical points.
I can justify why a maximum and minimum value must occur on a closed interval.
I can identify the critical points of a function by determining when the derivative is zero or undefined.
I can distinguish between relative and absolute extrema by comparing function values.
Quick Lesson Plan
Topic 5.2 prepares students for the First Derivative Test by first introducing the Extreme Value Theorem (EVT) for continuous functions on a closed interval. Other necessary vocabulary is integrated into the discussion of the graph representing Apple stock values: absolute (global) extrema, and relative (local) extrema, and critical points.
Most students will recall investigations into maxima and minima from their Precalculus courses. (AP Calculus now requires students to justify the location of these extrema through analysis of derivatives.) For this Topic, however, we are interested in naming these characteristics and clarifying that extrema will occur at critical values of x or the endpoints of a closed interval. The Extreme Value Theorem is likely new territory for our students, but is a logical extension of their prior work with continuous functions. We have allowed plenty of time for student practice (which should include both written and verbal communication of their solutions).
This content is often integrated into multi-step free response questions that utilize the First or Second Derivative Test and will appear on both the AB and BC forms of the AP Test.
This lesson generally flows smoothly, however, students may forget that critical values include domain values where the first derivative may be undefined, not just equal to zero. Emphasize that possibility. Additionally, remind students to consider endpoints of a closed interval (later called the Candidate Test) when looking for relative (local) or global (absolute) extrema.