IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
AA HL Optimisation
Optimise models with implicit constraints and endpoint possibilities.
Learning goal
Solve advanced optimisation problems, including constraints suited to implicit differentiation.
Syllabus link
AA HL: AHL 5.14 · Topic 5 Calculus.
Big idea
An implicit constraint can sometimes be differentiated directly, avoiding unnecessary algebraic rearrangement.
Key relationship
\(\frac{dQ}{dx}=0\quad\text{for interior candidates}\)
Core idea
An implicit constraint can sometimes be differentiated directly, avoiding unnecessary algebraic rearrangement.
Solve advanced optimisation problems, including constraints suited to implicit differentiation.
Key results and method
- Define the objective and constraint.
- Choose explicit substitution or implicit differentiation strategically.
- Find all feasible candidates.
- Classify/compare them and interpret the result.
Worked example
A point \((x,y)\) lies on \(x^2+4y^2=16\) in the first quadrant. Maximise \(P=xy\).
- Differentiate the constraint: \(2x+8y y\'=0\Rightarrow y\'=-x/(4y)\).
- \(P\'=y+xy\'=y-x^2/(4y)=0\).
- Combine with the constraint to find the candidate.
Practice
Try these questions before moving on.
- Optimise a quantity constrained by an ellipse.
- Compare an interior critical point with endpoints.
- Choose between explicit and implicit methods for a given constraint.
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