Optimisation uses calculus to find best values (maximise or minimise) of functions that model real problems:
maximise profit, minimise cost, maximise volume for given surface area, minimise material used, etc.
| Term / Concept | Definition / Short explanation |
|---|---|
| Objective function | The function to be optimised (e.g., Profit P(x), Cost C(x), Volume V(x)). |
| Constraint | Equation(s) that link variables (e.g., fixed surface area, budget, material length). Used to eliminate extra variables. |
| Feasible domain | The allowed x-values after constraints (often xโฅ0 or a closed interval). Endpoints must be checked. |
| Stationary point | Point where derivative = 0; candidate for local max/min. Classify using second derivative or sign test. |
๐ Key steps when solving an optimisation problem
- Interpret the word problem carefully โ define variables with units (e.g., x = radius in cm).
- Write the objective function to maximise/minimise in terms of the variables.
- Use constraints to eliminate extra variables so objective depends on a single variable.
- Determine the feasible domain (often x โฅ 0 or an interval).
- Differentiate, solve f'(x)=0 to get candidate points; evaluate endpoints if domain closed.
- Classify each candidate (f”(x) or first derivative sign test) and interpret with units and context.
๐ง Paper tip
- Show modelling steps clearly: variable definitions, equation derivation, elimination of extra variables, differentiation, solution, classification, interpretation with units.
- When a closed interval exists, always check endpoints as well as stationary points for absolute optima.
- If technology is allowed, present GDC output (equation, root finding) but still write short reasoning โ examiners expect a mix of technology and clear math reasoning.
๐ What each concept means in real life (important to state in answers)
- Objective function โ usually a measurable business/physical quantity (profit in currency, cost in currency, volume in m3, surface area in m2). Always write the units when you give the final answer.
- Constraints โ reflect physical limits: fixed material, fixed perimeter, budget, legal limits. Explain why a constraint is realistic (e.g., “we only have 20 m of fencing”).
- Feasible domain โ ensures answers make sense (negative dimensions are impossible); if the domain is closed, absolute max/min may occur at endpoints โ check them.
- Optimal point interpretation โ translate back: “radius = 5 cm gives maximum volume of 392 cm3” โ explain practical implications (e.g., cost saved, material reduction).
๐ Real-world connection
Packaging design: minimise surface area for a given volume to reduce material use and cost; manufacturers use optimisation to balance strength vs material cost. In economics, firms use optimisation to set production where marginal cost = marginal revenue to maximise profit.
๐ EE / Research ideas
Study the environmental impact of packaging design optimization โ balance material vs protection. Combine optimisation with a life-cycle analysis for an Extended Essay.
Optimisation โ Examination Questions
๐ Multiple Choice Questions
MCQ 1.
What is the correct role of a constraint in an optimisation problem?
- A. It defines the objective function
- B. It restricts the feasible values of variables
- C. It identifies the stationary point directly
- D. It replaces differentiation
Show Answer
Correct answer: B
Constraints represent physical or practical limits (such as fixed material or volume)
and are used to restrict the feasible domain and eliminate extra variables.
MCQ 2.
Why must endpoints be checked in optimisation problems with a closed domain?
- A. Endpoints are always stationary points
- B. Endpoints may give absolute maxima or minima
- C. Endpoints simplify differentiation
- D. Endpoints remove the need for constraints
Show Answer
Correct answer: B
When the feasible domain is closed, the maximum or minimum value
can occur at an endpoint rather than at a stationary point.
MCQ 3.
Which condition confirms a local maximum?
- A. fโฒ(x)=0 and fโณ(x)>0
- B. fโฒ(x)=0 and fโณ(x)<0
- C. fโฒ(x)>0
- D. fโณ(x)=0
Show Answer
Correct answer: B
A negative second derivative indicates the curve is concave down,
confirming a local maximum.
๐ Short Answer Questions
Question 1.
Why is it important to state units when interpreting an optimal solution?
Show Answer
Units give physical meaning to the result and show understanding of context.
For example, stating โx = 5โ alone is insufficient, but โx = 5 cmโ confirms
the solution is realistic and dimensionally correct.
Question 2.
Explain why optimisation problems are usually reduced to a single variable.
Show Answer
Differentiation can only be applied to functions of one variable at this level.
Constraints are therefore used to eliminate extra variables so that calculus
can be applied correctly.
๐ Long Answer Questions
Question 1.
A rectangular enclosure is built using 40 m of fencing on three sides
(the fourth side is a wall).
- Define suitable variables.
- Form the objective function for the area.
- Find the dimensions that maximise the area.
Show Full Solution
Step 1: Let x be the width (two sides) and y be the length.
Constraint: 2x + y = 40 โ y = 40 โ 2x.
Step 2: Area A = xยทy = x(40 โ 2x) = 40x โ 2x2.
Step 3: Differentiate: Aโฒ(x) = 40 โ 4x.
Set Aโฒ(x)=0 โ x = 10.
Step 4: y = 40 โ 2(10) = 20.
Conclusion: Maximum area occurs when width is 10 m and length is 20 m.
Question 2.
A cylindrical can must hold 1000 cm3 of liquid.
- Express height in terms of radius.
- Form the surface area function.
- Find the radius that minimises surface area.
Show Full Solution
Volume constraint: ฯr2h = 1000 โ h = 1000/(ฯr2).
Surface area S = 2ฯr2 + 2ฯrh.
Substitute h: S(r) = 2ฯr2 + 2000/r.
Differentiate: Sโฒ(r) = 4ฯr โ 2000/r2.
Set Sโฒ(r)=0 โ r3 = 500/ฯ.
This value gives the minimum material usage.
๐ฑ GDC Tips
- TI-Nspire CX II: Define the objective function in Graphs, use Menu โ Analyze โ Derivative โ Zero to locate stationary points.
- TI-Nspire CX II: Use the Table feature to check endpoint values when a domain is restricted.
- Casio fx-CG50 / CG100: Enter function in GRAPH, use G-Solv โ Min/Max to find optimal values.
- Casio fx-CG50 / CG100: Always verify constraints by checking domain limits manually.
๐ examiner notes
- Clarity wins marks: define variables, show elimination of extra variables, state feasible domain, show derivative algebra, and write clear final interpretation with units.
- Check endpoints and include brief discussion of practical constraints (manufacturing tolerances, minimum thickness, etc.).