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.
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.
Identify the quantity to optimize (maximize/minimize) as your objective function
List all given constraints on the variables in the problem
Rewrite the objective function as a function of a single variable using the constraints
Differentiate the single-variable function to find the first derivative
Set the first derivative equal to zero and solve for critical points
Check that the critical point is in the domain of the function, then classify it
Justify that the critical point is a global extremum (required for full marks)
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.
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Let = length of sides perpendicular to the house, = length parallel to the house. We want to maximize area .
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The total fencing constraint is: , so rearrange to get . Domain is .
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Substitute into the area formula to get a single-variable objective function:
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Differentiate and set equal to zero:
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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.
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Find . The maximum area dimensions are 25 m (perpendicular) Γ 50 m (parallel).
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