Overview
This lecture introduces the concepts of global and local (relative) maximum and minimum values in the context of functions, using the analogy of a roller coaster graph.
Global Maximum and Minimum
- The global (absolute) maximum is the highest point on the graph for the entire function.
- The global (absolute) minimum is the lowest point on the graph for the entire function.
- When talking about "max" or "min," it usually refers to these global values unless specified otherwise.
Local (Relative) Maximum and Minimum
- Local (relative) maxima and minima are the highest or lowest points within a small, specific interval around a point.
- If you zoom in on a section of the graph, a local max or min may appear which is not the global one.
- A point is a local max if it is higher than all nearby points; it's a local min if it is lower than all nearby points.
- For example, g(x) has relative minima at x = c and x = e, and relative maxima at x = b and x = f.
- The value at a local max/min depends on the interval you are considering (e.g., between a and c or between d and f).
Determining Maxima and Minima in Intervals
- A value is a relative max if it is greater than or equal to all other values on a specified interval.
- The same logic applies for relative min: it must be less than or equal to all values on the interval.
- Determining intervals is important for classifying points as relative extrema.
Key Terms & Definitions
- Global (Absolute) Maximum — The highest value of a function on its entire domain.
- Global (Absolute) Minimum — The lowest value of a function on its entire domain.
- Local (Relative) Maximum — The highest value of a function within a small interval around that point.
- Local (Relative) Minimum — The lowest value of a function within a small interval around that point.
- Interval — A specific subset of the domain used to define local extrema.
Action Items / Next Steps
- Review the graph to identify all local and global maxima and minima.
- Practice determining intervals that contain relative maxima or minima.