Solving Optimization Problems
Justify an optimum over the whole feasible domain and report the requested setting, output, and units.
College Board Topic 5.11 · CED p. 104 (PDF p. 109) · CED effective Fall 2020; project course outline 2026–27.
A finished optimization solution tells the reader what the optimum means. Distinguish the optimizing input, all derived dimensions, and the optimal output. “x = 3” is incomplete if x is a radius and the question asks for height and material area. State units and justify the result across the feasible domain, including endpoints or limiting behavior when appropriate.
For a continuous objective on a closed interval, compare all critical-point and endpoint values. For an open or unbounded interval, the EVT does not automatically apply: use complete derivative signs, boundary behavior, or a valid one-critical-point argument. A local turn seen in a calculator window is insufficient evidence of a global optimum. A constraint can also place the optimum on a boundary even when there is no interior stationary point.
Consider a closed cylinder with fixed volume 16π cm³. Volume gives h = 16/r² for r > 0. Surface area becomes S(r) = 2πr² + 32π/r. Then S′ = 4πr − 32π/r² = 4π(r³ − 8)/r², so the area decreases up to r = 2 and increases afterward. The minimum-area cylinder has radius 2 cm, height 4 cm, and surface area 24π cm². The evidence addresses the whole positive-radius domain, not merely a small neighborhood.
Let's work through it
For production amount q measured in modeled units, a cost is C(q) = 100 + q²/10 + 40/q for q > 0, measured in hundreds of dollars.
Why: The stated model and positive domain define the objective.
C′(q) = q/5 − 40/q² = (q³ − 200)/(5q²), so q = ∛200 is the only critical number.
Why: Solve the derivative equation without losing the positive-domain restriction.
C′ is negative before ∛200 and positive afterward.
Why: The numerator’s sign and positive denominator prove a global minimum.
The minimizing model output is q = ∛200; if actual output must be a whole number, compare neighboring feasible integers separately.
Why: A continuous optimum does not automatically solve a discrete production decision.
Try it from memory
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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
Why report units after optimizing?
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
Why report units after optimizing?
Worked answer and reason
They identify whether the answer is a dimension, cost, area, or another quantity.
An input value and an objective value often have different units.
Retrieval 2
Does a graph-window minimum prove the global optimum?
Worked answer and reason
No; justify the full domain and boundaries.
A window can omit a better feasible value.
Retrieval 3
If the modeled optimum is noninteger but items must be whole, what next?
Worked answer and reason
Compare the neighboring feasible integer choices using the actual objective, accounting for the domain.
Rounding the input alone can choose the wrong cost.
Catch the tempting mistake
- A local maximum always gives the largest feasible value.
- The global claim requires evidence over the full feasible domain.
- The best dimensions and the largest area are the same answer.
- Dimensions locate the design; area evaluates its objective.
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 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
- Use a three-part final-answer box: optimizing input, design/setting, objective value.
- For numerical roots, require a recorded interval and enough guard digits for the objective comparison.
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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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Extreme Value Theorem
The Extreme Value Theorem guarantees an absolute maximum and absolute minimum for a function continuous on a closed bounded interval. The extrema can occur at endpoints or interior points. For example, x² on [−1,2] attains minimum 0 at 0 and maximum 4 at 2. An open interval or a missing continuity hypothesis does not carry the same guarantee.