How do you reduce a two-variable expression to one variable?

Use the given constraint (e.g. fixed volume) to eliminate one variable.

How do you find the optimum value of the variable?

Differentiate and solve \frac{dy}{dx}=0.

How do you differentiate \frac{2000}{x}?

Write as 2000x^{-1}, giving -2000x^{-2}=-\frac{2000}{x^2}.

Why check the domain of x?

The stationary point must be a sensible value, e.g. a length that is positive and possible.

What must you give at the end of an optimisation question?

The value asked for, with units, in the context of the problem.

When might you compare end-point values?

When the quantity is optimised over a closed interval of the variable.

Optimisation problemsEdexcel International A Level Maths: Flashcards

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What is the first step in an optimisation problem?

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What is the first step in an optimisation problem?: Define the quantity to be maximised or minimised.
How do you reduce a two-variable expression to one variable?: Use the given constraint (e.g. fixed volume) to eliminate one variable.
How do you find the optimum value of the variable?: Differentiate and solve $\frac{dy}{dx}=0$ .
How do you show a stationary point is a maximum?: $\frac{d^2y}{dx^2}<0$ there.
How do you show it is a minimum?: $\frac{d^2y}{dx^2}>0$ there.
Fencing: 80 m on three sides, perpendicular sides $x$ . Area?: $A=x(80-2x)=80x-2x^2$ .
Open box, square base $x$ , volume $500$ : surface area?: $S=x^2+\frac{2000}{x}$ .
How do you differentiate $\frac{2000}{x}$ ?: Write as $2000x^{-1}$ , giving $-2000x^{-2}=-\frac{2000}{x^2}$ .
Why check the domain of $x$ ?: The stationary point must be a sensible value, e.g. a length that is positive and possible.
Closed cylinder, radius $r$ : surface area formula?: $S=2\pi r^2+2\pi rh$ .
What must you give at the end of an optimisation question?: The value asked for, with units, in the context of the problem.
When might you compare end-point values?: When the quantity is optimised over a closed interval of the variable.

Exam questions on Optimisation problems

A farmer uses 80 m of fencing to enclose a rectangular area against a straight wall. The wall forms one side, so fencing is needed on only three sides. The two sides perpendicular to the wall each have length $x$ metres, and the enclosed area is $A$ m $^2$ .
Show that this value of $x$ gives a maximum, and find the maximum area.2 marks
An open-topped box has a square base of side $x$ cm and height $h$ cm, and its volume is $500$ cm $^3$ . Its total outside surface area (the base and four sides) is $S$ cm $^2$ .
Given that $S$ has a minimum value at this stationary point, find the minimum value of $S$ .2 marks
A company sells $x$ hundred phone cases each day. Its daily profit, $P$ hundred pounds, is modelled by $P=-x^3+9x^2-15x-10$ , for $x\ge0$ .
Show that $P$ is stationary when $x=1$ and when $x=5$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).