3 6 Critical Points and Extrema What are

3. 6 Critical Points and Extrema What are extrema of a function and how do we find them?

Critical Points • Tangent - a line that intersects a curve in exactly one point. • Horizontal – left/right • Vertical – up/down • The three types of critical points occur when a tangent line is horizontal or vertical. • Maximum (plural Maxima) – when the graph changes from increasing to decreasing, point C • Minimum (plural Minima) – when a graph changes from decreasing to increasing, Point A • Point of Inflection (POI) – a point where the graph changes curvature, Point B POI

Absolute vs Relative • Absolute Max/min – there is not any point on the graph which is higher (max) or lower (min). • Relative Max/Min – the graph goes from increasing to decreasing (max) or decreasing to increasing (min) BUT it is only the high/low point RELATIVE to the points around it. The graph goes from increasing to decreasing, BUT there are other points that are higher The graph goes from decreasing to increasing, BUT there are other points that are lower

Example 1: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Example 1: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Example 1: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Example 2: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Example 2: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Example 2: Determine where the function has maxima/minima. Identify if they are absolute or relative. Write your answer in a complete sentence. •

Summary • NONE