Optimisation — Word Problems & AM-GM

1. Express the target $Q$ and the constraint. 2. Eliminate a variable so $Q=Q(t)$ in one variable. 3. Solve $Q'(t)=0$ for critical $t$. 4. Verify with $Q''$ or the first-derivative test (and check the domain's endpoints). Classic set-ups: a cylinder/box of fixed volume with least surface, the shortest distance from a point to a curve, $\sum (x-a_j)^2$ minimised at the mean.

Model the area/volume, use the geometric constraint (perimeter, radius, given side), reduce to one variable (often an angle $\theta$), and optimise. Useful facts: max-area triangle inscribed in a circle of radius $R$ is equilateral (area $\tfrac{3\sqrt3}{4}R^2$); fixed-perimeter rectangle ⇒ square; a sector of fixed perimeter has max area at a specific radius/angle.

For positive terms, AM $\ge$ GM: $\dfrac{u+v}{2}\ge\sqrt{uv}$, equality at $u=v$. So a sum with fixed product is minimised, and a product with fixed sum is maximised, when the terms are equal. Examples: $a^2x+b^2y$ with $xy=c^2$ has min $2abc$; $x+y=k\Rightarrow xy$ is max at $x=y=k/2$.