Introduction to Optimization Problems
Name an objective, constraint, and feasible domain before choosing an optimization procedure.
College Board Topic 5.10 · CED p. 103 (PDF p. 108) · CED effective Fall 2020; project course outline 2026–27.
Optimization begins before the derivative. Name the quantity you want to make large or small: that is the objective. Then identify the constraint and use it to rewrite the objective in one variable. State a feasible domain from the physical situation. A mathematically valid root outside that domain is not a possible design.
For a rectangular pen with 60 meters of fencing on all four sides, 2x + 2y = 60 gives y = 30 − x. Its area is A(x) = x(30 − x), with 0 < x < 30 for a nondegenerate pen. The derivative is A′ = 30 − 2x, yielding x = 15. The derivative changes from positive to negative, and the area tends to zero at both degenerate endpoints, so the square gives the largest area.
Other contexts share the same structure: cost under a production constraint, volume under material restrictions, or distance to a curve. If the objective is a nonnegative distance D, minimizing D² can be easier because squaring preserves the order of nonnegative quantities. Keep track of what you minimized so that the final answer reports a distance, not its square. A calculator can help solve a complicated derivative equation after a correct model exists.
Let's work through it
A rectangular pen uses 80 m of fencing on three sides against a wall. Let x be each side perpendicular to the wall and y the parallel side. Then 2x + y = 80.
Why: The unfenced wall side is not included in the material constraint.
A(x) = x(80 − 2x), with 0 < x < 40.
Why: The constraint eliminates y and the lengths must be positive.
A′ = 80 − 4x gives x = 20; A′ changes positive to negative there.
Why: A sign test proves this candidate maximizes the objective.
The parallel side is y = 40 m and maximum area is 800 m².
Why: Translate the maximizing input back into all requested physical quantities.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Which comes first: objective/constraint or differentiation?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Which comes first: objective/constraint or differentiation?
Worked answer and reason
Objective, constraint, and feasible domain.
Calculus cannot repair an incorrect physical model.
Retrieval 2
Why can squared distance be minimized instead of distance?
Worked answer and reason
Both are nonnegative and squaring preserves their order.
The same inputs minimize them, but the final units differ.
Catch the tempting mistake
- Any zero of the objective derivative solves the design problem.
- Check feasibility and justify maximum/minimum over the entire allowed domain.
- Use every visible side in a fencing constraint.
- Count only sides that actually need fencing in the stated context.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 5 problem 21 · No calculator · multiple
- Unit 5 problem 22 · No calculator · multiple
- Unit 5 problem 23 · No calculator · analytic
- Unit 5 problem 24 · No calculator · multiple
- Unit 5 problem 25 · Graphing calculator required · analytic
- Unit 5 problem 30 · No calculator · analytic
For the teacher
- Collect objective, constraint, and domain as a model-only checkpoint.
- Use a changed constraint, such as a wall or different side costs, to reveal whether learners are reasoning or recalling a square answer.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Optimization
Optimization finds the greatest or least value of an objective while satisfying the problem’s constraints. Express the objective in suitable variables, determine the allowed domain, and compare all relevant candidates. For example, among rectangles with a fixed perimeter, writing area in terms of one side reduces the search to a one-variable problem. Endpoint and feasibility checks are part of the argument, not optional final details.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Constraint in optimization
A constraint restricts the quantities or choices permitted in a problem. It can be an equation, an inequality, or a requirement from the physical context. For example, a rectangle with perimeter 20 must satisfy 2l+2w=20 with positive side lengths. Use constraints to determine the objective’s allowed domain before searching for extrema.