Optimization

Math

Definition

The process of finding the maximum or minimum value of a function, often subject to constraints. In calculus, this involves finding critical points where the derivative equals zero or is undefined, then testing which gives the extreme value.

How It Works

  1. Identify the quantity to maximize or minimize and write it as a function
  2. Express the function in terms of one variable using given constraints
  3. Take the derivative and set it equal to zero to find critical points
  4. Use the second derivative test or endpoints to confirm max/min
  5. Answer the original question with correct units

Examples

  • Finding the dimensions of a box with maximum volume given a fixed amount of material
  • Minimizing the cost of fencing to enclose a given area
  • Finding the point on a curve closest to a given point
Key Fact

At a maximum or minimum, f'(x) = 0 or f'(x) is undefined

Related Terms

Extreme values Related rates Derivative definition

Study This Concept

Practice optimization with free review games in these units:

AP Calculus AB Unit 5: Analytical Applications of Differentiation