A calculus optimization workflow that does not skip the model
Turn a word problem into one objective function, then use calculus and endpoint checks to justify the best feasible value.
LaTeX article Updated July 11, 2026
Objective versus constraint
The objective is the quantity being optimized: area, cost, distance, or time. Constraints describe what combinations are allowed. Mixing them produces an equation with too many variables and no clear target.
Use the constraint to eliminate variables from the objective before differentiating.
The feasible domain matters
Physical lengths cannot be negative, capacities may be bounded, and denominators may exclude values. The domain is part of the model and determines which critical points are candidates.
On a closed interval, endpoints must be compared with interior critical points.
Justify the optimum
A zero derivative identifies a stationary candidate, not automatically a maximum. Use sign changes, the second derivative, or direct value comparison to justify the conclusion.
State the requested quantity in context rather than ending with an unexplained variable value.
Worked example
Common mistakes
- Differentiating the constraint instead of the objective.
- Keeping two independent variables after a constraint is available.
- Ignoring endpoints or feasibility.
Keep these ideas
- Name the objective explicitly.
- Reduce to one variable before differentiating.
- A critical point needs a maximum or minimum justification.