Optimization

Here's calculus earning its keep. "What's the largest, cheapest, fastest, or smallest?" is a question about a maximum or minimum — and you already know those live where the derivative is zero.

The best point has a flat tangent

At a maximum or minimum, the curve stops rising and hasn't yet started falling (or vice versa), so its tangent is momentarily flat: the derivative is zero. Optimization is the art of writing the quantity you care about as a function, then solving f′ = 0.

The recipe

1. Name the quantity to optimize and write it as a function.
2. Reduce to one variable, using any constraint to eliminate the others.
3. Differentiate and set to zero to find the critical points.
4. Confirm it's a max or min (second-derivative test or endpoints), and answer the question asked.

Worked example

You have 40 m of fencing for a rectangular pen against no wall. Maximize the area.

• Perimeter constraint: 2x + 2y = 40, so y = 20 − x.
• Area as one variable: A(x) = x(20 − x) = 20x − x².
• Differentiate and solve: A′(x) = 20 − 2x = 0 → x = 10.
• Then y = 10, and A″(x) = −2 < 0 confirms a maximum.

The best pen is a 10 × 10 square with area 100 m² — the graph above peaks exactly there.

Why this matters

Optimization is the most directly useful thing calculus does: minimizing material in packaging, maximizing range on a tank of fuel, finding the least-cost production level. Every one reduces to the same move — set the derivative to zero.