5.11 Solving Optimization Problems

Step-by-step shortcut you can use on every AP optimization problem: 1. Read context → define the objective function (what you maximize/minimize) in one variable. If it’s given with two variables, use the constraint to eliminate one (solve for y in terms of x). 2. Identify the feasible region (domain or physical bounds from the problem). Endpoints matter on closed intervals. 3. Differentiate: compute f′(x). Find critical points where f′(x)=0 or undefined that lie in the feasible region. 4. Classify candidates: use the Second Derivative Test (f″(x)>0 → local min, <0 → local max) or the First Derivative Test. For closed intervals, evaluate f at all critical points and endpoints—the largest/smallest value is the global max/min. 5. Interpret answer in context with units and constraints (FUN-4.C). If there’s a constraint harder to eliminate, use Lagrange multipliers (BC or tougher AB contexts). Practice these steps on problems (CED-style)—see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and hit practice questions (https://library.fiveable.me/practice/ap-calculus).

Finding a maximum vs a minimum in optimization is the same process but different interpretation and final check. You: - set up an objective function and constraint(s) to get the feasible region, - find critical points where f′(x)=0 or undefined, and then evaluate candidates (critical points and endpoints/boundary). To decide type: - use the First Derivative Test or Second Derivative Test at each critical point to classify local max vs local min, - use the Candidates Test (compare f at all critical points and endpoints) to pick the global (absolute) maximum or minimum on the feasible region. In applied problems always interpret units (FUN-4.C) and check endpoints/boundary conditions—an endpoint can be the global best even if interior points give local extrema. On the AP exam you must justify your choice (show derivative work and comparison), and explicitly state what the extremum means in context. For review, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7), Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5), and practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the first derivative test when you want to determine whether a critical point (where f′ = 0 or f′ is undefined) is a local max, local min, or neither by checking sign changes of f′ around that point. It works whenever you can evaluate f′ on either side—and it still works if f′ or f″ is messy or undefined. Use the second derivative test when f′(c) = 0 and f″ exists near c: compute f″(c). If f″(c) > 0, f has a local minimum at c; if f″(c) < 0, a local maximum. If f″(c) = 0 or f″ doesn’t exist, the second derivative test is inconclusive—go back to the first derivative test. Always remember AP requirements: check all candidates (critical points and endpoints/boundary) when finding absolute extrema (Candidates Test). Interpret your result in context (FUN-4.C). For a concise refresher and practice problems on optimization, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5).

Start by identifying what quantity the problem asks you to minimize or maximize—that is your objective (or target) function. Next list the constraints: equations or inequalities that link the problem’s variables (these define the feasible region). Use the constraints to eliminate extra variables so your objective is written as a single-variable function. Quick checklist (CED-aligned): - Pick the objective (area, volume, cost, distance, etc.). - Write constraint equations from geometry or wording. - Solve the constraints for one variable and substitute into the objective to get f(x) of one variable. - Find critical points: f′(x)=0 or f′ undefined, plus check boundary/endpoint values (Candidates Test). - Use first/second derivative tests to classify local vs global extrema and then interpret the result in context (units, feasibility). Remember to always interpret your extremum in the real context (FUN-4.C): discard nonphysical solutions and report units. For worked examples and step-by-step practice aligned with AP Topic 5.11, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7). For extra practice problems, check Fiveable’s practice bank (https://library.fiveable.me/practice/ap-calculus).

The constraint equation comes from the fixed relationship(s) in the word problem—what cannot change. Steps to find it: 1. Define variables for the things that can change (e.g., let x = length, y = width). 2. Read the problem for a fixed total (perimeter, area, volume, budget, material, etc.). Translate that into an equation relating your variables—that’s the constraint. - Example: “fixed perimeter 100” → 2x + 2y = 100. - Example: “fixed volume V of a box” → V = x y h. 3. Use the constraint to eliminate one variable from the objective function (the quantity to maximize/minimize). That gives a single-variable function to differentiate. 4. Determine the feasible region (domain) from physical restrictions (x>0, integer, etc.) and check endpoints + critical points for global extrema. 5. Interpret your result in context (units and meaning), per FUN-4.C in the CED. For examples and practice tied to Topic 5.11, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and try problems at https://library.fiveable.me/practice/ap-calculus.

When a problem asks for the “minimum cost” or “maximum area” it means: build a mathematical model (an objective function) whose value measures cost or area, subject to the problem’s constraints (the feasible region). Your job is to find the input(s) (dimensions, time, number of items, etc.) in that feasible region that make the objective as small (minimum cost) or as large (maximum area) as possible, and then interpret that value with correct units. How you do it (AP-aligned): - Write the objective function and eliminate extra variables using the constraint(s). - Find critical points where f′(x)=0 or f′ is undefined, and check endpoints/boundaries of the feasible region (candidates test). - Use the first/second derivative tests or compare values to identify global min/max. - State the final answer in context (e.g., “minimum cost = $120 when length = 3 m”), including units and why it’s the global extreme (critical point vs. endpoint). This interpretation step—connecting the computed min/max back to the real-world meaning—is exactly the FUN-4.C skill on the AP CED. For worked steps and examples, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

A max or min isn’t just a number—it’s the best/worst value of your objective in the real situation. Always say three things: (1) what variable gets that value, (2) the objective’s value with units, and (3) whether it’s global (absolute) or local and why (critical point or endpoint). Example: “The profit is maximized at x = 120 widgets; maximum profit = $3,600,” tells the reader the production level and the money you get. In constrained problems, clarify the feasible region: the optimum must satisfy constraints (e.g., rope length, budget). Use critical points (f′ = 0 or undefined) and the Candidates Test (include endpoints) to find absolute extrema; use first/second derivative tests to classify local extrema. On the AP exam you must interpret with units and context—don’t just give the x-value. For worked strategies and AP-style practice, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For more practice problems, try (https://library.fiveable.me/practice/ap-calculus).

You check endpoints because the absolute (global) max or min on a closed feasible region can occur there—not just at critical (stationary) points where f′(x)=0 or undefined. The Candidates Test (used on the AP exam) requires you to evaluate all critical points AND the boundary points (endpoints or other boundary conditions from the constraint) and compare f-values to decide the global extremum. In applied problems the endpoint often has real meaning (smallest/ largest possible dimension, time limit, physical constraint), so skipping it can give the wrong answer. So: (1) find critical points inside the feasible interval, (2) evaluate the objective at those points and at the endpoints/boundary, (3) pick the largest/smallest value. This matches FUN-4.C (interpret minima/maxima in context) and the Candidates Test in Topic 5.11. For extra practice, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).

Check the answer against the real situation in these quick steps—they map to the CED keywords (feasible region, endpoints, interpretation of min/max, derivative tests): 1. Units and meaning: make sure your result has the right units and that you can state it in words (e.g., “minimum cost = $120 when width = 3 m”). That’s FUN-4.C: interpret the value in context. 2. Feasible region and constraints: confirm your critical point lies inside the allowed domain (not negative, within physical limits). If it’s outside, the real answer is an endpoint. 3. Endpoint/boundary check: evaluate the objective at endpoints and any boundary cases. The global min/max could be there (candidates test). 4. First/second derivative check: use f′ = 0 or undefined to find candidates, use f″ or first-derivative sign to classify local vs global extrema. 5. Magnitude sanity check: is the number reasonable? Compare to rough estimates or limits (order-of-magnitude). If it’s absurd, revisit model/formula. 6. Re-plug into original context: compute any dependent quantities (area, cost, time) and explain what they mean in words. For practice doing this on AP-style problems and to see worked examples that stress interpretation, check the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Start by naming variables and writing the objective function (what you want to maximize or minimize) and the constraint(s) (equation(s) linking the variables). 1) Variables → objective f(x, y, ...) (e.g., area, volume, cost). 2) Constraint(s) → g(x, y, ...) = 0 (e.g., perimeter fixed, budget). 3) Reduce to one variable: solve the constraint for one variable and substitute into the objective to get a single-variable function F(x). - Formulaic setup: maximize/minimize F(x) = f(x, y(x)) subject to domain from the feasible region. 4) Find critical points: solve F′(x) = 0 and check endpoints of the feasible interval (Candidates Test). 5) Classify extrema: use F″(x) (second-derivative test) or first-derivative sign analysis, and interpret in context (units, feasible region). For multi-variable constraints, use Lagrange multipliers: solve ∇f = λ∇g together with g = 0, then check feasible points and boundary values. Always justify why your candidate gives a global min/max on the feasible region (CED keywords: objective function, constraint, feasible region, critical point, candidates test, Lagrange multipliers). More practice and worked examples are in the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For lots of practice questions, see (https://library.fiveable.me/practice/ap-calculus).

Related rates and optimization both use derivatives, but they answer different questions. - Related rates: You’re finding how one quantity’s rate of change depends on another’s (use dy/dt, dx/dt). Set up an equation relating the quantities (constraint), differentiate implicitly with respect to time, then plug given rates. Typical CED context: units, interpretation of derivative. Example: how fast is the top of a ladder sliding down when the bottom moves away? - Optimization: You’re finding extrema (min or max) of an objective function subject to constraints. Build an objective function, reduce it to one variable using the constraint, find critical points (f′ = 0 or undefined), test with first/second derivative or candidates test (include endpoints/boundary), and interpret in context (FUN-4.C keywords: objective function, feasible region, global minimum/maximum). Use related rates when the problem gives rates and asks for another rate. Use optimization when it asks for the largest/smallest value (area, cost, time, etc.). For AP-style practice and steps, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Start by naming the objective (what you want to max/min) and the constraint (what’s fixed). Use geometry formulas to write the objective as a single-variable function, find critical points, and test them (first/second derivative or the Candidates Test), then check endpoints/boundary of the feasible region. Steps (rectangles and cylinders): - Rectangle with fixed perimeter P: let width = x, then height = h = (P/2) − x. Area A(x) = x((P/2) − x) = (P/2)x − x^2. Compute A′(x)= (P/2) − 2x, set =0 → x = P/4 (square). Use A″(x)= −2 to confirm max, and check endpoints if domain limited. - Cylinder with fixed volume V (minimize surface area): S(r,h)=2πr^2+2πrh, constraint V=πr^2h ⇒ h = V/(πr^2). Substitute: S(r)=2πr^2 + 2V/r. Differentiate: S′(r)=4πr − 2V/r^2, set =0, solve for r; use S″(r)>0 to confirm min, check physical domain (r>0). Remember to interpret units and context (FUN-4.C). For more worked examples and AP-style practice, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and try problems at (https://library.fiveable.me/practice/ap-calculus).

1) Read the problem and identify what you’re maximizing or minimizing (the objective function) and any constraint(s). Translate the context into an equation for the objective in terms of one variable if possible. 2) Use the constraint(s) to eliminate extra variables so your objective is a single-variable function on a feasible domain (feasible region / boundary conditions). Check domain endpoints from the context. 3) Differentiate: find f ′(x) and solve f ′(x) = 0 to get critical (stationary) points inside the feasible region. Also note where f ′ is undefined if those points lie in the domain. 4) Test candidates: evaluate the objective at all critical points and at endpoints/boundary points. Use the First Derivative Test or the Second Derivative Test (f ″(x) > 0 local min, < 0 local max) as needed to classify local extrema. For constrained multivariable problems, consider Lagrange multipliers. 5) Pick the global max/min by comparing values and interpret the answer in context (units, what the number means). State any assumptions/limitations. For AP-style applied problems, show work for each step and include endpoint analysis; see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7) and unit review (https://library.fiveable.me/ap-calculus/unit-5). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: whether an optimization problem has zero, one, or many solutions depends on the objective function and the feasible region (constraints). Use the Candidates/Extreme Value tests from the CED: check critical points, endpoints/boundary, and global behavior. Why no solution - Feasible region is empty (constraints impossible). - Domain is open or unbounded and the objective is unbounded (e.g., maximize x on (0, ∞)—no maximum). - Function isn’t defined or isn’t continuous on the region so EVT doesn’t apply. Why multiple solutions - A flat region/plateau: f is constant on an interval or along a boundary so every point there attains the same extremum. - Symmetry or multiple distinct critical points give equal objective values (e.g., symmetric geometry problems). - Multiple boundary points tie for the best value. What you should always do on AP problems: identify the feasible region, find critical points, analyze endpoints/boundary, test behavior at infinity, and use first/second-derivative or EVT reasoning to justify existence/uniqueness. For a focused review, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7); for more practice, try the unit page (https://library.fiveable.me/ap-calculus/unit-5) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Write your answer like a short story that proves you followed the CED steps and interprets the result in context. Checklist (use on every optimization FRQ) - Define variables in words and with units. (e.g., “let x be the width in meters.”) - Write the objective function and any constraint(s). Show algebra that reduces to one-variable objective. - State the feasible region (domain/constraints) before differentiating. - Find critical points: compute f′(x), set = 0 (and check where f′ undefined). Show algebra or calculus steps. - Test candidates: evaluate objective at critical points AND at endpoints/boundary. Use the Candidates Test (or 2nd-derivative/1st-derivative test to justify a local extremum). - Give a clear conclusion sentence: identify the global max or min, give the value with units, and restate what the value means in context (FUN-4.C style). Example final sentence style: “The volume is maximized when x = 3 m; the maximum volume is 54 m^3.” Tip: if your calculus shows a local extremum, explicitly compare values to show it’s global. For more worked examples and phrasing, see the Topic 5.11 study guide (https://library.fiveable.me/ap-calculus/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7). For broader Unit 5 review and extra practice, check the unit page (https://library.fiveable.me/ap-calculus/unit-5) and the practice bank (https://library.fiveable.me/practice/ap-calculus).