Optimization with a Constraint Equation

Most optimization problems hand you a quantity to maximize or minimize, like volume or area, that depends on two variables. You can't take a derivative with respect to two variables at once using single-variable calculus, so the constraint equation, some fixed relationship those variables must obey, does half the work. It lets you solve for one variable in terms of the other and substitute, collapsing the problem down to a single-variable function. Only then does d/dx mean anything.

Once you have one function of one variable, the rest is Unit 5 as usual: take the derivative, set it to zero, check it's a max using the first or second derivative test, and confirm the critical point actually sits inside the domain the problem allows. That last check matters more here than anywhere else, because the formula you built often stays defined for x values that make no physical sense, like a negative length or a cutout bigger than the sheet. The real domain comes from the situation, not the algebra.

This topic is where Unit 5's theory becomes a tool. Finding critical points and classifying them with derivative tests, from earlier in the unit, only becomes useful for real problems once you can reduce a two-variable setup to one variable. The next step after this is usually optimization without a tidy algebraic constraint, where the relationship between variables is implicit, but the core move, substitute, differentiate, check the domain, confirm the extremum, stays exactly the same.