Topic 5.10 Introduction to Optimization Problems: set up an optimization problem by writing the quantity to be optimized as a function of one variable.

What this topic is asking

The College Board (Topic 5.10) is the setup half of optimization: translating a word problem into an objective function of one variable with a domain. Topic 5.11 then solves it. You must identify the quantity to optimize, use the constraint to eliminate variables, and state the domain.

The setup procedure

A worked setup

Why the domain matters

The domain is not a formality. Optimization problems are usually solved by the candidates test on the closed interval (or by checking the single interior critical point), and that needs the correct endpoints. A box problem with $0 < x < 6$ has its maximum at an interior critical point, but many problems have their extremum at an endpoint of the realistic domain, and you cannot find it without stating the domain first. The physical restrictions, all lengths positive, a fixed total of material, set the domain, so read them off the problem before reducing variables.

The constraint is what makes it one variable

The defining feature of an optimization problem is the constraint that links the variables, letting you eliminate all but one. Without a constraint, a two-variable objective like area $A = xy$ has no single maximum. The constraint (a fixed perimeter, a fixed volume of material) supplies the second equation, and substituting it collapses the objective to one variable. Identifying the constraint correctly is the step students most often get wrong: misreading which quantity is fixed leads to the wrong objective function and a wrong answer despite correct calculus later. Slow down on the constraint, and the rest follows.

A reliable checklist for the setup

A short checklist turns the setup into a routine. First, draw and label the situation, naming the variable dimensions; a diagram prevents confusion about which length is which. Second, write the objective as an equation, even if it starts in two variables. Third, write the constraint as a separate equation. Fourth, solve the constraint for one variable and substitute into the objective to get a single-variable function. Fifth, read off the domain from the physical limits. Working through these five steps in order, rather than trying to leap straight to a one-variable formula, catches the common errors of mislabelling, picking the wrong constraint, or forgetting the domain. The discipline matters because the later calculus is only as good as the function it is applied to.

Why this is examined separately from solving

The College Board splits optimization into setup (this topic) and solving (the next) because the setup carries distinct, scoreable skills: translating a context into an objective and a constraint, eliminating a variable, and stating a domain. On a free-response question these earn points on their own, and a correct setup with a small later error still scores well. Conversely, flawless differentiation applied to a wrong objective earns little. Investing time to get the one-variable objective and its domain right is therefore the highest-value part of an optimization problem, which is exactly why it has its own topic.