AP Calculus AB and BC glossary

Optimization

Also called: Optimisation

In an optimization problem you are handed a quantity to make as large or as small as possible, usually with a restriction on the variables.

The objective is the quantity being made large or small. The constraint is the relationship the variables are forced to obey. Most problems open with two equations, one of each, and the constraint is what lets you rewrite the objective in a single variable. When the objective already arrives in one variable, as it does whenever a model such as $V(x)$ is handed to you outright, there is no constraint equation to solve and only the domain to respect.

A candidate is any input that could produce the extreme value. That means every critical number inside the domain, which is a point where the derivative is zero or where it fails to exist, together with each endpoint when the domain is closed. Comparing the objective's values across that whole list is what the phrase candidates test names. For the step-by-step routine, from writing the objective to writing the justification, see the guide on optimization word problems.

The mistake

Which inputs count as candidates? Not only the solutions of $A' = 0$ . An interior point where $A'$ fails to exist is a critical number too, and on a closed domain both endpoints join the list. A second habit worth breaking is answering with the input that maximises a quantity when the question asked for the quantity itself.

Appears in: Unit 5: Analytical Applications