Optimization uses derivatives to locate maxima or minima. Express the objective in one variable, find critical points from derivative zero or undefined points, and test candidates with endpoints or sign analysis.
- Translate words into objective and constraint equations first.
- Reduce objective to one variable before differentiating.
- Compare all valid candidates, not only where derivative is zero.
Set up objective and constraint cleanly
Most optimization errors happen before calculus starts. Define the quantity to maximize or minimize, then use the constraint to eliminate one variable.
Domain limits matter. Physical dimensions cannot be negative and often have upper bounds.
Find and test critical candidates
Differentiate the one-variable objective and solve for critical points. Then test with first derivative sign changes, second derivative, or direct comparison including endpoints.
Report the final result with units and the variable values that produce it.
- State feasible interval clearly.
- Check endpoint values for closed intervals.
- Reject critical points outside the domain.
Maximize rectangle area with fixed perimeter
Perimeter is 40 m. Let length be x and width be y.
- Constraint: 2x + 2y = 40, so y = 20 - x.
- Objective: A(x) = x(20 - x) = 20x - x^2, with 0 < x < 20.
- Derivative A'(x) = 20 - 2x = 0 gives x = 10, then y = 10.
Common mistakes
- Differentiating before removing extra variables.
- Ignoring endpoint checks when interval is closed.
- Accepting algebraic critical points outside feasible domain.
- Reporting x value without objective value and units.
Try one
Why is checking endpoints required in many maximum or minimum problems?
Because absolute extrema on a closed interval can occur at endpoints, not only at critical points.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.