Optimisation problemsEdexcel International A Level Maths: Mind map

Exam questions on Optimisation problems

1. 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
2. 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
3. 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