Chapter 10 — Key Takeaways
The one-sentence version: Real-world optimization is a translation problem — turn English into one function of one variable using the constraint, restrict to the feasible domain, find critical points, then prove which one is the global extremum. The calculus is mechanical; the setup is the craft (§10.1).
1. The Standard Procedure (English → Calculus)
The eight steps of §10.3, compressed. Follow them in order and optimization stops being mysterious.
- Draw and label. Name the variables on a picture. Most geometry problems become obvious once drawn.
- Write the objective — the quantity to maximize or minimize (area, cost, time, profit, rate), possibly in several variables at first.
- Write the constraint — the fixed relationship linking the variables (perimeter, volume, demand).
- Eliminate — use the constraint to reduce the objective to a function of one variable.
- Find the feasible domain of that variable (lengths $\ge 0$, a cut $\le$ half the sheet, etc.).
- Find critical points — where the derivative is zero or undefined inside the domain.
- Classify and confirm the global extremum (see §3 below).
- Translate back — report the original quantity with units and a plausibility check.
Steps 1–4 are setup; 5–8 are calculus and bookkeeping.
2. Objective vs. Constraint — Don't Confuse Them
This is the single most important distinction in the chapter (§10.2).
| Objective | Constraint | |
|---|---|---|
| Role | The thing you optimize; you differentiate it | The thing you substitute to eliminate variables |
| Tell | The quantity you are free to push | The quantity that is fixed |
| Examples | "maximize area," "minimize material," "maximize profit" | "100 ft of fencing," "holds 350 mL," "demand is 18,000/yr" |
Confuse them and you optimize the wrong thing. The constraint is what collapses a multivariable objective down to one variable — without it, you cannot proceed.
3. Verifying the Global Extremum (the step students skip)
A critical point is a candidate, never an answer (§10.13). Match the domain to the right confirmation tool (§10.15):
| Domain | Tool | What you do |
|---|---|---|
| Closed, bounded $[a,b]$ | Closed-interval method (§10.4) | Evaluate $f$ at every critical point and both endpoints; largest = global max, smallest = global min. Gold standard — cannot fail. |
| Open / unbounded $(a,\infty)$ etc. | Boundary-limit argument (§10.6) | If $f \to +\infty$ at both ends, the lone interior critical point is the global min (and mirror for max). |
| Single critical point | Sole-critical-point rule | A continuous function with one critical point that is a local max has it as the global max (and likewise min). |
| Constant concavity | Convexity argument | $f'' > 0$ everywhere ⟹ convex ⟹ any critical point is the global min; $f'' < 0$ ⟹ global max. |
Make the global-extremum justification an explicit written step, not an afterthought.
4. The Closed-Interval Method (the workhorse)
When $f$ is continuous on $[a,b]$, the Extreme Value Theorem (Ch. 9) guarantees a global max and min exist, and they live only at (i) interior critical points or (ii) endpoints — nowhere else (§10.4). So:
- Find all critical points in $(a,b)$.
- Evaluate $f$ at each, plus at $a$ and $b$.
- Compare the finite list.
No second-derivative test required. This method cannot mistake a local extremum for a global one — its great virtue.
5. Why a Maximum Forces $f' = 0$
At an interior extremum where $f$ is differentiable, $f'(c) = 0$ — Fermat's theorem (§10.4, Ch. 9). The peak of a hill and the bottom of a valley are both flat. But the theorem says nothing about endpoints (only a one-sided slope exists) or points where $f'$ is undefined — which is exactly why the candidate list has three kinds of point, not one.
6. The Recurring Results Worth Remembering
These clean structural facts recur and are worth carrying:
- Pen against a wall (§10.5): with fixed fencing $L$, the parallel side is twice each perpendicular side; max area $= L^2/8$.
- Free rectangle, fixed perimeter: the square maximizes area.
- Minimum-material closed can (§10.6): height = diameter ($h = 2r$).
- Closest point on a curve (§10.7): minimize $D^2$, not $D$ — same minimizer, polynomial derivative. Geometrically, the shortest segment meets the curve perpendicularly.
- Least-time path (§10.8): $\dfrac{\sin\theta_1}{v_1} = \dfrac{\sin\theta_2}{v_2}$ — Snell's law; light minimizes time automatically (Fermat's principle).
- Profit max (§10.10): marginal revenue = marginal cost, i.e. $R'(x) = C'(x)$, because $P' = R' - C' = 0$.
- Economic Order Quantity (§10.11): $Q^* = \sqrt{\dfrac{2DK}{h}}$; at the optimum, ordering cost = holding cost.
- Optimal foraging (§10.12): marginal value theorem — leave a patch when $g'(t^*) = \dfrac{g(t^*)}{\tau + t^*}$ (instantaneous rate falls to the long-run average).
- Maximum power transfer (§10.14): load resistance = source resistance ($R = \rho$).
7. Common Errors to Avoid
- Skipping the endpoints on a closed interval (§10.4). The maximum of $x^3 - 3x$ on $[0,3]$ is at the endpoint, not the interior critical point. Always evaluate $a$ and $b$.
- Assuming a critical point is the extremum you wanted (§10.13). For fixed perimeter the square minimizes the diagonal; the longest diagonal lives at the degenerate boundary. Read what you actually computed.
- Differentiating $\sqrt{\,\cdot\,}$ directly in distance problems (§10.7). Minimize the square instead.
- Confusing objective and constraint (§10.2) — optimizing the fixed quantity instead of the free one.
- Building the wrong objective (§10.6, §10.12). The minimum-material can is taller in reality because the real objective includes lid thickness, seams, and shipping. "Garbage in, optimum out."
- Reporting a local extremum as global with no justification (§10.15).
8. Connections — Backward and Forward
- Builds on: the Extreme Value Theorem and Fermat's theorem (Ch. 9), the first- and second-derivative tests (Ch. 9), and all the differentiation rules (Ch. 7), especially the quotient rule (§10.14) and chain rule (§10.8).
- Single-variable only. Every problem here had one free variable after the constraint. When variables genuinely can't be reduced to one — minimizing a box's cost where length, width, and height all trade off — you need the multivariable tools.
- Looks ahead to Chapter 11: Newton's method uses the derivative to solve equations (including the critical-point equation $f'(x) = 0$ when it has no closed form, as in Exercise D2-style cubics).
- Lagrange multipliers and the multivariable second-derivative (Hessian) test live in Chapter 31. There, a critical point is where $\nabla f = \mathbf{0}$, and "optimize on the road" with an unsolvable constraint becomes a system of equations. The perpendicularity of §10.7 reappears as parallel gradients.
- The gradient-descent anchor (introduced Ch. 6, full ML form Ch. 30) is this chapter's principle scaled to thousands of variables: descend until the derivative (gradient) is zero.
The deepest lesson. Optimization is where calculus stops describing and starts deciding. A farmer's fence, a can's shape, a lifeguard's route, a firm's output, a bird's feeding schedule — all bend to one three-line procedure: write the objective, use the constraint to reach one variable, set the derivative to zero and check the boundary. The calculus is the easy part. Judging what to optimize, what is fixed, and what is plausible — that is the craft you carry into every quantitative field.