Applied Extrema

Finding the Best Value: Applied Extrema

Optimization problems ask: what is the maximum area, minimum cost, or optimal design given real-world constraints? The strategy is straightforward: set up two equations—an objective function (the quantity being optimized) and a constraint (a fixed relationship between variables)—then reduce to a single-variable problem solvable by calculus.

The Core Method

Every optimization has two essential equations:

Solve the constraint for one variable and substitute into $Q$ to eliminate variables. This produces $Q(x)$, a function of a single variable ready for critical-point analysis on a domain determined by physical meaning (e.g., $x > 0$).

Worked Example: Maximizing Rectangular Area

A farmer has 100 meters of fencing. Maximize the enclosed area. Let sides be $x$ and $y$.

Constraint: $2x + 2y = 100 \Rightarrow y = 50 - x$

Objective: $Q(x) = xy = x(50-x) = 50x - x^2$ for $0 < x < 50$

Optimize: $Q'(x) = 50 - 2x = 0 \Rightarrow x = 25$. Since $Q''(x) = -2 < 0$, this is a maximum. The optimal field is a $25 \times 25$ square with area $625 \text{ m}^2$.

Key Pitfalls to Avoid

Forgetting the domain: Physical constraints (lengths positive, volumes positive) must restrict $x$. In the farming example, $x \in (0, 50)$, not all real numbers. Endpoints represent degenerate shapes and yield zero or nonsensical values.

Skipping reasonableness checks: Always verify that your critical point makes physical sense. Negative dimensions or impossible values signal setup errors.

The Optimization Checklist

1. Identify objective quantity $Q$ in terms of two (or more) variables
2. Write the constraint equation linking those variables
3. Solve constraint for one variable; substitute into $Q$
4. Determine domain from physical restrictions
5. Find critical points using $Q'(x) = 0$; apply derivative test or closed-interval method
6. Verify the answer is physically reasonable

Mastering steps 1–2—translating a word problem into correct equations—is the genuine skill. Once $Q(x)$ is established on its proper domain, calculus delivers the answer.