Optimization

Optimization uses constraints, derivatives, and endpoint comparisons to find the best feasible value of a model.

Critical points and concavity changes organize the shape of a differentiable graph. A smooth curve has a local maximum, local minimum, and inflection point marked. Sign bands below the axis show intervals of increase and decrease, while curvature visibly changes near the inflection point.

Optimization turns “largest,” “smallest,” “least costly,” or “most efficient” into a function over a realistic domain.

Objective and constraint

The objective function is the quantity to maximize or minimize. A constraint connects the variables and restricts what is feasible.

The goal is usually to rewrite the objective using one independent variable, then analyze it on the domain allowed by the problem.

Worked example: maximum rectangle area

Closed-interval method

If the objective is continuous on $[a,b]$:

find critical numbers inside $(a,b)$;
evaluate the objective at those numbers and at $a,b$;
compare outputs.

The largest is the absolute maximum and the smallest the absolute minimum on the interval.

Minimum-cost example structure

Suppose total cost is

C(x)=5000+20x+\frac{80000}{x},qquad x>0.

Differentiate:

C'(x)=20-\frac{80000}{x^2}.

Solving $C'=0$ gives a candidate; the second derivative or a sign chart can classify it. Context may require rounding to a whole production batch and comparing nearby feasible integers.

Geometry formulas and units

Optimization often uses perimeter, area, surface area, or volume. Write units throughout:

length: units;
area: square units;
volume: cubic units.

Dimensional consistency can reveal that the wrong quantity was optimized.

Common mistakes

Optimizing the constraint instead of the objective. Identify what the question asks to maximize or minimize.

Keeping multiple independent variables. Use the constraint to reduce the model.

Ignoring endpoints or physical restrictions. The optimum may occur at a boundary.

Assuming every critical number is the required optimum. Classify and compare.

Rounding a discrete answer without checking neighbours. Compare feasible candidates.