Lecture 4.1: Maxima and Minima

Importance of Maxima and Minima

• Objective: Determine the largest or smallest value of a quantity in practical problems.
• Approach: Translate practical problems into mathematical descriptions to find maximum or minimum of functions.

Applications of Derivatives

• Chapter 4 Focus: Use derivatives in applied settings as a tool.
• Previous Knowledge: Chapter 3 discussed finding derivatives of different functions.
• Current Focus: Applications and terminologies related to optimization.

Terminology

• Global Maximum: Largest y-value over the entire segment of the curve.
• Global Minimum: Smallest y-value over the entire segment of the curve.
• Local (or Relative) Maximum: Largest y-value within a small region around a point.
• Local (or Relative) Minimum: Smallest y-value within a small region around a point.
• Extrema: Collective term for maximum and minimum values of a function.

Analogy and Mathematical Concepts

• Analogy: Consider tallest and shortest people in a dorm by floor and in the entire dorm.
• Mathematical: Global considers all y-values, local considers y-values within a small region.

Examples

• Piecewise Function: Label global and local extrema using graphs.
• Endpoint Limitation: Local extrema cannot occur at endpoints due to the need for comparison with neighbors.

Theorem and Critical Points

• Theorem: If a function attains a max/min at x = c, then derivative at c (f'(c)) is either 0 or does not exist.
• Critical Points: Points where the derivative is zero or does not exist; candidates for max/min.
  • Not all critical points are extrema; further analysis is needed.

Extreme Value Theorem

• Statement: If f is continuous on [a, b], there exist minimum and maximum y-values on that interval.
• Application: Determine global extrema by evaluating endpoints and critical points within the interval.

Example Problems

1. Function g(x) = 4x/(x²+1) on [0, 2]

   • Calculate values at endpoints and critical points.
   • Determine global extrema using values derived from those calculations.

2. Function f(x) = ln(x²+2) on [-1, 2]

   • Use derivative to find critical points.
   • Evaluate function at critical points and endpoints to find global extrema.