Solving Optimization Problems

Every optimization problem hides two equations: one for the quantity you want to optimize, and one constraint linking the variables. Start by naming variables and writing the objective (area, cost, distance) in terms of more than one variable if needed. Then use the constraint to eliminate all but one variable, so you have a single-variable function to differentiate. Take the derivative, set it to zero, and solve for the critical points. This step only gives candidates. The actual answer depends on the shape of the domain: if the domain is a closed interval, you must also evaluate the function at the endpoints and compare, since the true minimum or maximum can sit there instead of at the critical point. If the domain is all real numbers or an open interval, use the first or second derivative test to confirm the critical point is a genuine max or min, not just a flat spot.

A can's shape depends on the cost of materials because the objective function changes with the problem. If the lid and base cost more per square cm than the side, the surface area formula gets weighted terms, and the optimal radius shifts to use less of the expensive part. The math only reflects whatever costs you build into the function.

This topic is where the whole unit pays off: the shape-reading tools from earlier sections (critical points, first and second derivative tests, concavity) become a method for answering real questions about the best possible value, not just describing a curve.