Math
Optimization.
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 · 5 phases
Step by step.
- Identify the quantity to maximize or minimize and write it as a function
- Express the function in terms of one variable using given constraints
- Take the derivative and set it equal to zero to find critical points
- Use the second derivative test or endpoints to confirm max/min
- Answer the original question with correct units
Examples
Real-world.
- 1 Finding the dimensions of a box with maximum volume given a fixed amount of material
- 2 Minimizing the cost of fencing to enclose a given area
- 3 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