Optimisation (HL contexts)
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Flip to reveal answersWhat are the four steps of an optimisation problem?
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Question
What are the four steps of an optimisation problem?
Answer
1) Write the quantity. 2) Reduce to one variable using the constraint. 3) Differentiate and set = 0. 4) Justify max/min and answer the question.
Question
What condition holds at the optimum value?
Answer
The first derivative is zero (a stationary point).
Question
How do you justify a stationary point is a maximum?
Answer
Second-derivative test: f″ < 0 ⇒ maximum (or f′ changes + to − through the point).
Question
How do you justify a stationary point is a minimum?
Answer
f″ > 0 ⇒ minimum (or f′ changes − to + through the point).
Question
Why minimise D² instead of D for a shortest-distance problem?
Answer
D² has the same minimising x as D (it's a monotone transformation) but avoids the messy square-root derivative.
Question
Open box from an a×a sheet (corner squares x): volume formula?
Answer
V = x(a − 2x)², with domain 0 < x < a/2.
Question
Why must you check the domain / reject some solutions?
Answer
Lengths can't be negative and some roots make a dimension zero — those are physically impossible.
Question
After finding the optimum variable, what is often still required?
Answer
The optimum VALUE — substitute the variable back into the original quantity.
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Full study notes for Optimisation (HL contexts)
Topic 5.14 hub
Implicit & related rates (HL only)
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