Concept preview

What you will be able to do

1. 01Carry out the full optimization procedure: define variables, write the constraint, reduce the objective to a function of ONE variable, and set a sensible domain.
2. 02Locate candidate points by solving objective$'=0$, check the domain endpoints, and justify that a candidate is a maximum or minimum with the first or second derivative test.
3. 03Report the exact quantity the problem asks for, with correct units — avoiding the classic trap of reporting the optimizing variable instead of the requested value.

An optimization problem hands you two things tangled together: a quantity to make as large or as small as possible (the OBJECTIVE), and a restriction that ties your variables together (the CONSTRAINT). A rectangle has both an area and a perimeter; if the perimeter is fixed, the area is what you push to a maximum. The whole method is a disciplined way of separating these two roles and then applying single-variable calculus.