Study Guide

Optimization applications

IB Mathematics: Analysis and Approaches SLΒ· 15 min read

1. The General Optimization Method

All optimization problems follow a consistent framework that helps you avoid errors and earn full method marks in IB exams.

πŸ“˜ Definition

Optimization Problem

A problem that requires finding the maximum or minimum value of an objective function, subject to one or more constraints that limit the possible values of the problem variables.

  1. Identify the quantity to optimize (maximize/minimize) as your objective function

  2. List all given constraints on the variables in the problem

  3. Rewrite the objective function as a function of a single variable using the constraints

  4. Differentiate the single-variable function to find the first derivative

  5. Set the first derivative equal to zero and solve for critical points

  6. Check that the critical point is in the domain of the function, then classify it

  7. Justify that the critical point is a global extremum (required for full marks)

πŸ“ Worked Example

A rectangular garden is built alongside a house, so no fencing is needed along the house. You have 100 m of fencing total. Find the dimensions that give the maximum area of the garden.

  1. 1

    Let = length of sides perpendicular to the house, = length parallel to the house. We want to maximize area .

  2. 2

    The total fencing constraint is: , so rearrange to get . Domain is .

  3. 3

    Substitute into the area formula to get a single-variable objective function:

  4. 4
    A(x)=x(100βˆ’2x)=100xβˆ’2x2A(x) = x(100 - 2x) = 100x - 2x^2
  5. 5

    Differentiate and set equal to zero:

  6. 6
    dAdx=100βˆ’4x=0β€…β€ŠβŸΉβ€…β€Šx=25\frac{dA}{dx} = 100 - 4x = 0 \implies x = 25
  7. 7

    Classify and justify: The second derivative is , so this is a local maximum. As it is the only critical point in the domain, it is the global maximum.

  8. 8

    Find . The maximum area dimensions are 25 m (perpendicular) Γ— 50 m (parallel).

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When this came up on past exams

AI-estimated based on syllabus patterns β€” cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2022 Β· 1

    Maximum area of a fenced garden

  • 2021 Β· 2

    Minimum production cost

  • 2023 Β· 1

    Maximum volume of an open box

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