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{{T}}
[[Category:English]]
[[Category:Optimisation Problems]]

{{BR}}
= Introduction =

Optimisation problems ask you to find the '''best possible value''' of a quantity while respecting given conditions. In Grades 11–13, this usually means turning a real situation into a mathematical model, identifying a quantity to maximise or minimise, and using [[English:Calculus|calculus]] to justify the optimum. Typical goals include maximising area or volume, minimising cost or distance, and choosing dimensions that use material efficiently.

By the end of this aiMOOC, you should be able to identify decision variables, objective functions and constraints; determine a feasible domain; locate and classify candidate extrema; compare endpoints and critical points; interpret an optimum in context; and explain why your answer is mathematically and practically reasonable.

[[File:Maxima and Minima.svg|600px|frameless|center]]

The graph above illustrates why an optimisation problem may contain both '''local''' and '''global''' extrema. A local maximum is higher than nearby values, while a global maximum is at least as high as every feasible value in the whole domain. Optimisation normally asks for a global optimum unless the context says otherwise.

{{#ev:youtube|https://www.youtube.com/watch?v=lx8RcYcYVuU|500|center}}

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= What Is an Optimisation Problem? =

An optimisation model has three central ingredients. A '''decision variable''' represents a quantity you are free to choose. An '''objective function''' gives the quantity you want to maximise or minimise. '''Constraints''' describe the restrictions that your choice must satisfy. The values satisfying all constraints form the '''feasible set'''.

If the feasible set is an interval or other domain <math>D</math> and the objective is <math>f</math>, then a point <math>c\in D</math> is a global maximum when <math>f(c)\geq f(x)</math> for every <math>x\in D</math>. It is a global minimum when <math>f(c)\leq f(x)</math> for every feasible <math>x</math>.

[[File:ConvexFunction.svg|500px|frameless|center]]

If a strictly convex function on a convex feasible interval attains a minimum, that minimiser is unique. Convexity becomes increasingly important in advanced [[English:Mathematical optimization|mathematical optimisation]].

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== From Context to Mathematics ==

A strong solution begins before you differentiate. Read the situation carefully and decide what can vary. Draw a labelled diagram when geometry is involved. Define variables with units. Write the quantity to be optimised. Use the constraints to express that quantity using as few independent variables as possible. Finally, determine the feasible domain from physical and mathematical restrictions.

For example, if a rectangle has perimeter 80 m and one side is <math>x</math> metres, the other side is <math>40-x</math>. Its area is therefore <math>A(x)=x(40-x)</math>, and the physically meaningful domain is <math>0\leq x\leq 40</math>. The constraint has reduced a two-variable problem to a one-variable objective.

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= Calculus Tools for Optimisation =

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== Derivatives and Critical Points ==

The [[English:Derivative|derivative]] measures instantaneous rate of change. At a differentiable interior local maximum or minimum, the derivative must be zero. Such points are called stationary points. More generally, a '''critical point''' is an interior point where the derivative is zero or does not exist.

[[File:Derivative with tangent.svg|500px|frameless|center]]

A horizontal tangent is an important signal, but it is not a guarantee of an optimum. For example, <math>f(x)=x^3</math> has <math>f'(0)=0</math>, yet <math>x=0</math> is neither a local maximum nor a local minimum. You must classify critical points and compare all relevant candidates.

{{#ev:youtube|https://www.youtube.com/watch?v=S0_qX4VJhMQ|500|center}}

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== First Derivative Test ==

The first derivative test classifies a critical point by checking how the sign of <math>f'</math> changes around it. If <math>f'</math> changes from positive to negative, the function changes from increasing to decreasing and has a local maximum. If <math>f'</math> changes from negative to positive, the point is a local minimum. If there is no sign change, the point is not a local extremum.

This method is useful when the derivative factors neatly or when you can make a sign chart. It also gives you information about the increasing and decreasing behaviour of the whole function.

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== Second Derivative Test ==

If <math>f'(c)=0</math> and the second derivative exists, then <math>f''(c)>0</math> indicates a local minimum and <math>f''(c)<0</math> indicates a local maximum. When <math>f''(c)=0</math>, the test is inconclusive; another method is required.

[[File:Parabola2.svg|400px|frameless|center]]

The graph of <math>y=x^2</math> has positive curvature and a minimum at the origin. This is the geometric idea behind the second derivative test for a stationary point with positive second derivative.

{{#ev:youtube|https://www.youtube.com/watch?v=-cW5hCsc9Yc|500|center}}

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== Global Extrema on a Closed Interval ==

The [[English:Extreme value theorem|Extreme value theorem]] states that a continuous function on a closed, bounded interval has both an absolute maximum and an absolute minimum. To find them, evaluate the objective at every interior critical point and at both endpoints, then compare the resulting values. If the function is not differentiable at an interior point, that point must also be considered.

Endpoints matter because the optimum can occur at the boundary. In applied problems, the boundary may represent a limiting design such as zero height, full capacity, a minimum legal value, or the largest permitted resource use.

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= A Reliable Solution Strategy =

Use the following workflow for one-variable optimisation problems:
# [[English:Mathematical modelling|Mathematical modelling]]: Define the decision variable clearly and include units.
# [[English:Objective function|Objective function]]: Write the quantity that must be maximised or minimised.
# [[English:Constraint (mathematics)|Constraint]]: Use the given relationship to reduce the objective to one independent variable when possible.
# [[English:Domain of a function|Feasible domain]]: State all allowed values before differentiating.
# [[English:Critical point|Critical point]]: Differentiate and solve for all relevant critical points.
# [[English:Extremum|Classification]]: Use a sign test, second derivative test, or direct comparison.
# [[English:Boundary value problem|Boundary check]]: Evaluate feasible endpoints or limiting boundary cases.
# [[English:Mathematical proof|Justification]]: Compare candidates and explain why the chosen value is globally optimal in the stated domain.
# [[English:Dimensional analysis|Interpretation]]: Give the final result with units and check that it makes sense in context.

{{#ev:youtube|https://www.youtube.com/watch?v=Ef22yTJDUZI|500|center}}

{{BR}}
= Worked Example: Maximum Rectangle with Fixed Perimeter =

A rectangle has perimeter 80 m. What dimensions maximise its area?

Let one side be <math>x</math> metres. The perimeter condition gives the other side as <math>40-x</math>, so
<math display="block">A(x)=x(40-x)=40x-x^2,\qquad 0\leq x\leq 40.</math>

Differentiate:
<math display="block">A'(x)=40-2x.</math>

The stationary point satisfies <math>40-2x=0</math>, so <math>x=20</math>. Also, <math>A''(x)=-2<0</math>, so the stationary point is a local maximum. The endpoints both give area zero, while <math>A(20)=400</math>. Therefore the global maximum area is '''400 m²''', achieved by a '''20 m by 20 m square'''.

This example shows an important structural result: among all rectangles with a fixed perimeter, the square has maximum area.

{{BR}}
= Worked Example: Open-Top Box =

A 24-inch square sheet of cardboard has equal squares of side <math>x</math> cut from each corner. The sides are folded up to make an open-top box. What cut size maximises the volume?

The box has height <math>x</math> and square base side length <math>24-2x</math>. Thus
<math display="block">V(x)=x(24-2x)^2,\qquad 0\leq x\leq 12.</math>

Expanding and differentiating gives
<math display="block">V'(x)=576-192x+12x^2=12(x-4)(x-12).</math>

The interior critical point is <math>x=4</math>. At the endpoints <math>x=0</math> and <math>x=12</math>, the volume is zero. At <math>x=4</math>, the volume is
<math display="block">V(4)=4\cdot16^2=1024.</math>

Therefore the maximum volume is '''1024 cubic inches''', obtained by cutting out '''4-inch squares'''. Notice that the feasible domain was essential: the algebraic root <math>x=12</math> lies at a degenerate boundary where the base disappears.

{{#ev:youtube|https://www.youtube.com/watch?v=3GYv-BZYYdg|500|center}}

{{BR}}
= Worked Example: Minimum Material for a Closed Cylinder =

Suppose a closed cylindrical container must hold a fixed volume <math>K</math>. Let its radius be <math>r</math> and its height be <math>h</math>. The volume constraint is <math>\pi r^2h=K</math>, while the surface area to minimise is <math>S=2\pi r^2+2\pi rh</math>.

From the constraint, <math>h=K/(\pi r^2)</math>. Substitution produces a one-variable objective:
<math display="block">S(r)=2\pi r^2+\frac{2K}{r},\qquad r>0.</math>

Then
<math display="block">S'(r)=4\pi r-\frac{2K}{r^2}.</math>

Setting this equal to zero gives <math>2\pi r^3=K</math>. Since <math>S''(r)=4\pi+4K/r^3>0</math> for <math>r>0</math>, this critical point is the unique global minimum. Substituting the critical-point relation back into the height formula yields <math>h=2r</math>. Therefore the least-surface-area closed cylinder has '''height equal to its diameter'''. This example shows how a constraint can be used to eliminate one variable before applying calculus.

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= Constraints, Feasibility and Alternative Methods =

Not every optimisation problem is best solved by differentiation. When both the objective and constraints are linear, [[English:Linear programming|linear programming]] can be used. In two variables, the constraints define a feasible region, and a linear objective—when an optimum exists on a bounded polygonal feasible region—reaches an optimum at a vertex.

[[File:Linear Programming Feasible Region.svg|450px|frameless|center]]

This graphical method highlights the same three ideas used in calculus optimisation: an objective, constraints and a feasible set. The technique changes, but modelling remains central.

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== Beyond One Variable ==

Some advanced problems depend on several independent variables. Then partial derivatives, the [[English:Gradient|gradient]] and methods such as [[English:Lagrange multiplier|Lagrange multipliers]] can replace ordinary one-variable differentiation. For Grades 11–13, the main goal is to understand the modelling logic and recognise how one-variable ideas extend to higher dimensions.

[[File:Paraboloid-KSE.svg|500px|frameless|center]]

A bowl-shaped surface gives a geometric picture of a two-variable minimum. In higher-dimensional optimisation, level curves, gradients and constraint surfaces help describe the direction in which an objective changes most rapidly.

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= Technology as a Checking Tool =

A graphing calculator, computer algebra system or spreadsheet can help you inspect a function, estimate a maximum or minimum, test sensitivity to parameters, and check arithmetic. Technology is most useful after you have defined the model correctly.

Do not treat a graphing window as a proof. A poor viewing range can hide extrema, numerical solvers can miss solutions, and a calculator may return a critical point outside the feasible domain. Your written solution should still identify the objective, constraints, domain, critical points and reasoning used to select the optimum.

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= Common Errors and How to Avoid Them =

# [[English:Domain of a function|Ignoring the domain]]: A mathematically valid critical point may be impossible in the real situation.
# [[English:Stationary point|Assuming every stationary point is optimal]]: A zero derivative can also occur at a non-extremal point.
# [[English:Endpoint|Forgetting endpoints]]: A global optimum on a closed interval can occur at a boundary.
# [[English:Mathematical modelling|Optimising the wrong quantity]]: Check that your formula actually represents the requested area, volume, cost, distance or profit.
# [[English:Units of measurement|Dropping units]]: State the unit of every variable and of the final objective value.
# [[English:Rounding|Rounding too early]]: Keep exact values during algebra when possible and round only at the end.
# [[English:Verification and validation|Skipping interpretation]]: Substitute your result back into the original constraints and ask whether it is realistic.

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= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''What is the objective function in an optimisation problem?'''
(The quantity to maximise or minimise)
(!The list of all units)
(!A graphing calculator setting)
(!The name of the decision variable)

{{E}}
<br>

{{MC}}

'''Why must you state a feasible domain before differentiating?'''
(It identifies the values allowed by the problem)
(!It guarantees every derivative is positive)
(!It removes the need to test endpoints)
(!It makes every function linear)

{{E}}
<br>

{{MC}}

'''What does a change in the first derivative from positive to negative indicate?'''
(A local maximum)
(!A local minimum)
(!A vertical asymptote)
(!A missing constraint)

{{E}}
<br>

{{MC}}

'''What does a change in the first derivative from negative to positive indicate?'''
(A local minimum)
(!A local maximum)
(!A constant function)
(!An impossible domain)

{{E}}
<br>

{{MC}}

'''What does a positive second derivative at a stationary point indicate?'''
(A local minimum)
(!A local maximum)
(!A guaranteed endpoint)
(!A linear objective)

{{E}}
<br>

{{MC}}

'''What should you do when the second derivative is zero at a stationary point?'''
(Use another classification method)
(!Declare a maximum immediately)
(!Declare a minimum immediately)
(!Ignore the point)

{{E}}
<br>

{{MC}}

'''Which candidates must be compared for a continuous function on a closed interval?'''
(Critical points and endpoints)
(!Only the midpoint)
(!Only positive values)
(!Only stationary points)

{{E}}
<br>

{{MC}}

'''Why is substitution useful in constrained optimisation?'''
(It can reduce the number of independent variables)
(!It always removes the objective function)
(!It makes all constraints disappear)
(!It proves every point is feasible)

{{E}}
<br>

{{MC}}

'''Which method is especially suitable when both objective and constraints are linear?'''
(Linear programming)
(!Synthetic division)
(!Long multiplication)
(!Angle bisection)

{{E}}
<br>

{{MC}}

'''What is a good final check after finding an optimum?'''
(Interpret it in the original context)
(!Delete the units)
(!Ignore the constraints)
(!Round every value to an integer)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Objective function || Quantity to be maximised or minimised
|-
| Decision variable || Quantity that can be chosen
|-
| Constraint || Restriction that feasible choices must satisfy
|-
| Critical point || Interior point where the derivative is zero or undefined
|-
| Global optimum || Best value over the entire feasible domain
|-
| Endpoint || Boundary value that may need comparison
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Define a decision variable'''
| Choose a measurable quantity that can vary
|-
| '''Build the objective'''
| Express the quantity to maximise or minimise
|-
| '''State the feasible domain'''
| Restrict the variable to values allowed by the context
|-
| '''Find critical points'''
| Solve where the derivative is zero or undefined
|-
| '''Compare candidates'''
| Evaluate possible optima and select the best feasible value
|}
{{E}}

<br>
...
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Derivative || What rate-of-change tool is central to calculus optimisation?
|-
| Constraint || What restriction limits the feasible choices?
|-
| Maximum || What do you call the greatest feasible objective value?
|-
| Minimum || What do you call the least feasible objective value?
|-
| Endpoint || What boundary value of an interval must often be checked?
|-
| Feasible || What adjective describes a value that satisfies every constraint?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Optimisation+Problems </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
An optimisation problem begins by defining a { decision variable } that represents what you can choose. The quantity you want to maximise or minimise is the { objective function }. Restrictions on possible choices are called { constraints }. All allowed values together form the { feasible domain }. Interior points where a derivative is zero or undefined are called { critical points }. A sign change in the first derivative from positive to negative indicates a { local maximum }. On a closed interval, you must also compare the { endpoints }. A positive second derivative at a stationary point supports classification as a { local minimum }. Substitution can reduce a constrained problem to { one variable }. The final numerical result should always be interpreted in its original { context }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

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=== Easy ===
# [[English:Functions|Functions]]: Choose a simple quadratic function, draw its graph by hand or with graphing software, mark its optimum, and explain how the graph and derivative agree.
# [[English:Geometry|Geometry]]: Design two rectangles with the same perimeter, calculate their areas, and create a labelled image showing which design uses the perimeter more efficiently.
# [[English:Derivative|Derivative]]: Record a short video in which you explain why a horizontal tangent can be a clue to an optimum but does not always prove one.
# [[English:Mathematical modelling|Mathematical modelling]]: Interview someone about a real decision involving cost, time, space or materials, then identify a possible decision variable, objective and constraint.

{{BR}}
=== Standard ===
# [[English:Spreadsheet|Spreadsheet]]: Build a spreadsheet for the open-top box problem, vary the cut size, calculate the volume, create a graph, and compare the numerical maximum with the calculus result.
# [[English:Economics|Economics]]: Create a simple revenue, cost or profit model with stated assumptions, find an optimum, and write a short interpretation for a business audience.
# [[English:Physics|Physics]]: Investigate a motion or energy situation in which a maximum or minimum is meaningful, derive a suitable function, and explain the physical restrictions on its domain.
# [[English:Linear programming|Linear programming]]: Visit a school workshop, garden, kitchen, shop or other suitable place, identify two limited resources, and sketch a two-variable feasible region for a realistic planning problem.

{{BR}}
=== Advanced ===
# [[English:Engineering mathematics|Engineering mathematics]]: Design a container subject to a fixed-capacity constraint, optimise one measure of material use, and produce a scale drawing or 3D model of your proposed design.
# [[English:Data analysis|Data analysis]]: Collect or generate data near a predicted optimum, fit a simple model, and discuss how measurement noise could change your confidence in the best value.
# [[English:Calculus|Calculus]]: Compare the first derivative test, second derivative test and direct value comparison on three different functions, then write a decision guide for choosing a method.
# [[English:Operations research|Operations research]]: Produce a five-minute explanatory video that contrasts calculus optimisation with linear programming and gives one realistic use case for each.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Model construction|Model construction]]: Given an unfamiliar real-world scenario, define variables, formulate an objective and constraints, and justify the feasible domain before performing any calculus.
# [[English:Method selection|Method selection]]: Compare two optimisation problems and explain why one is best solved with derivatives while the other is better suited to linear programming or direct comparison.
# [[English:Error analysis|Error analysis]]: Analyse a worked solution that checks only where the derivative is zero, identify what information is missing, and determine whether the claimed optimum is valid.
# [[English:Parameter sensitivity|Parameter sensitivity]]: Change one parameter in a geometric optimisation model, predict how the optimum should change, and verify your prediction algebraically or numerically.
# [[English:Proof and interpretation|Proof and interpretation]]: Solve a constrained optimisation problem, prove that your candidate is global within the feasible domain, and interpret the result with correct units.
# [[English:Transfer task|Transfer task]]: Create an optimisation model from a context outside mathematics class, state its assumptions and limitations, solve it, and evaluate whether the mathematical optimum would be practical in real life.

<br>
<br>

{{BR}}
= Evidence of Learning =

# '''Knowledge''': You can explain objective functions, decision variables, constraints, feasible domains, critical points, endpoints, local extrema and global extrema.
# '''Skills''': You can translate a context into a model, differentiate an objective, classify candidates, compare boundary values, use technology critically and communicate a justified conclusion.
# '''Products''': Your evidence can include labelled diagrams, solved models, graphs, spreadsheets, short reports, presentations, videos or physical prototypes.
# '''Transfer''': You can recognise optimisation structures in geometry, economics, physics, engineering and planning, then select an appropriate mathematical method rather than applying differentiation automatically.
# '''Quality of reasoning''': Your final work states assumptions, preserves units, respects constraints, checks alternative candidates and explains the practical meaning and limitations of the optimum.

<br>
<br>

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= OERs on the Topic =

<iframe> https://en.m.wikipedia.org/wiki/Mathematical_optimization </iframe>

<br>

{{BR}}
= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Optimisation Problems|Optimisation Problems]]'''
# [[English:Calculus|Calculus]]
# [[English:Derivative|Derivative]]
# [[English:Extremum|Extremum]]
# [[English:Critical point|Critical point]]
# [[English:Mathematical modelling|Mathematical modelling]]
# [[English:Geometry|Geometry]]
# [[English:Linear programming|Linear programming]]
# [[English:Convex function|Convex function]]
# [[English:Economics|Economics]]
# [[English:Engineering mathematics|Engineering mathematics]]
|}

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Optimisation Problems]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Mathematical optimization]]
[[Category:Grades 11-13]]
[[Category:Secondary education]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Optimisation Problems - Versionsgeschichte

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