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.
How to recognize the method, run it, and know when it is the wrong choice.
Reviewed July 11, 2026Identify the quantity to maximize or minimize, use constraints to write it in one variable, determine the feasible interval, then compare critical points and endpoints.
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.
A rectangle has perimeter 40. Which dimensions maximize its area?
Use the perimeter constraint.
Write area in one variable.
Find the interior critical point.
Common mistakes
- Differentiating the constraint instead of the objective.
- Keeping two independent variables after a constraint is available.
- Ignoring endpoints or feasibility.
Three takeaways
- Name the objective explicitly.
- Reduce to one variable before differentiating.
- A critical point needs a maximum or minimum justification.