Extreme values

Math

Definition

Extreme values are the maximum and minimum values of a function on a given interval. The Extreme Value Theorem states that a continuous function on a closed interval [a, b] must attain both an absolute maximum and an absolute minimum.

How It Works

  1. Find the derivative and set it equal to zero to find critical points
  2. Evaluate the function at each critical point within the interval
  3. Evaluate the function at the endpoints a and b
  4. The largest value is the absolute maximum; the smallest is the absolute minimum

Examples

  • Finding the highest and lowest points on a roller coaster modeled by a function
  • Determining the maximum profit given a revenue function on a fixed domain
  • f(x) = x² on [-2, 3] has a minimum of 0 at x = 0 and a maximum of 9 at x = 3

Related Terms

Optimization Mean value theorem Continuity Antiderivatives

Study This Concept

Practice extreme values with free review games in these units:

AP Calculus AB Unit 5: Analytical Applications of Differentiation