Calculus 12 · Mathematics · Lesson 4.2
Optimization
- Level
- Grade 12
- Reading time
- 14 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
Max and min problems.
How do you solve optimization problems in calculus?
Short answer
Optimization finds the maximum or minimum value of a quantity. You express the quantity as a function of one variable, differentiate it, and set the derivative equal to zero to find critical points, since extrema occur where the slope is zero (or at endpoints). The second derivative or a sign test confirms whether each critical point is a maximum or a minimum.
Concept preview
What you will be able to do
- 01Interpret an optimization problem as finding the largest or smallest value of a quantity.
- 02Determine the critical points of a function where its derivative is zero or undefined.
- 03Apply the first-derivative test and the closed-interval method to classify maxima and minima.
- 04Solve an applied optimization problem with a systematic procedure, including the domain.
- 05Diagnose a flawed optimization argument, such as one that ignores endpoints or the domain.
A great many practical questions are really requests for a largest or smallest value: the dimensions of a box that hold the most for the least cardboard, the price that earns the greatest profit, the shape of a container that uses the least metal, the path that takes the least time. Problems of this kind are called optimization problems, and the derivative is the tool that solves them, because a smooth interior extremum requires a zero derivative, while corners and endpoints are additional candidates that must also be checked. Learning to translate a worded situation into a function and then to locate that function's highest or lowest value is one of the most useful things the whole of differential calculus can do.
School Plan
Optimization
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Related topics
Frequently asked questions
- Why set the derivative to zero?
- At a maximum or minimum the tangent line is horizontal, so the instantaneous rate of change — the derivative — is zero there.
- How do you know if a critical point is a max or a min?
- Use the second derivative test: negative means a maximum, positive means a minimum; or check the sign of the first derivative on each side.