Simple Optimization Problems

From "find the extrema of $f$" to "find the best real-world outcome." Many practical situations -- maximizing the area enclosed by a fixed length of fence, minimizing the material used to build a container of a given capacity, maximizing profit or minimizing cost -- reduce to exactly the maxima/minima problem of Sections 4 and 5, once the quantity to be optimized is written as a function of a single variable using whatever constraint the problem supplies.

Worked pattern: maximizing area under a fixed perimeter. For a rectangle of perimeter $P$, writing the two adjacent sides as $x$ and $\left(\tfrac{P}{2}-x\right)$, the area $A(x) = x\left(\tfrac{P}{2}-x\right)$ is a downward-opening parabola in $x$, maximized at its unique critical point $x = P/4$ -- giving equal sides, i.e. the maximizing rectangle is always a square (Exercise: Maxima and Minima, Q4, with $P = 40\ \text{m}$, giving a $10\ \text{m}\times10\ \text{m}$ square).

Worked pattern: maximizing volume of a folded box. Cutting a square of side $x$ from each corner of a flat sheet of side $a$ and folding up the flaps gives an open box of base side $(a - 2x)$ and height $x$, so $V(x) = x(a-2x)^2$; differentiating and solving $V'(x)=0$ locates the optimal cut size (Exercise Q5, $a = 18\ \text{cm}$, giving $x = 3\ \text{cm}$ -- note $x = a/2 = 9$ is also a root of $V'$, but it lies at the domain's edge, where the box collapses to zero volume, so it is rejected in favour of the interior critical point). …