Calculus: Critical Points and Local Extrema

Objectives

• Identify critical points using the definition
• Identify local maxima and minima using the definition
• Understand that local maxima and minima must occur at critical points but that not every critical point is the location of a local maximum or local minimum

Vocabulary

• critical point
• local maximum, minimum, extrema

About the Lesson

This lesson involves visualizing the connections between the critical points and local extrema. As a result, students will:

• Zoom in on function graphs at different types of critical points (including stationary points, locations of vertical tangents, “corners,” and cusps) to determine whether the slope of the tangent line is zero or undefined.
• See that a local maximum or minimum occurs at critical points, but the examples illustrate that not every critical point is a local extremum.
• Use the first derivative test as a means to identify local maximum and local minimum.
• Build on their familiarity with the concept of the derivative at a point as the local slope of the function graph at that point.