Introduction to Optimization Problems AB & BC

An optimization problem carries an objective, the quantity to be made largest or smallest, and a constraint, the relationship that limits the choices. The question decides which is which. For a rectangular plot with 1200 feet of fence on three sides, the objective is $A = xy$ and the constraint is $2x + y = 1200$; solving the constraint for $y$ and substituting gives $A(x) = x(1200 - 2x)$, a single function of a single variable that the rest of the unit can handle.

Two habits break the setup. The constraint goes unused, and $A = xy$ is differentiated with $y$ treated as a constant, which asserts that lengthening one side leaves the other alone. And the domain is taken from the algebra rather than from the situation: $V(x) = x(12 - 2x)^{2}$ is a polynomial defined everywhere and describes a real box only for $0 < x < 6$. Positivity, stated caps, whole-number counts, and the point where a formula stops applying are all part of the domain. When the objective is a distance, minimize its square, since both are smallest at the same input.